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Bad Wolffor other, similarly-named pages.
The "Bad Wolf" was an entity that Rose Tyler briefly became after staring into the heart of the Doctor's TARDIS and directly into the Time Vortex. The Bad Wolf scattered her name throughout time as a sign to herself that she was linked to the Ninth Doctor, creating a looped ontological paradox.
Biography
After looking into the Heart of the TARDIS, newly empowered, Rose Tyler piloted the TARDIS back to the Ninth Doctor on the Game Station. Interrupting the Daleks' plans to exterminate the Doctor, Rose, as the Bad Wolf, explained what she had done and then spread the words "Bad Wolf" throughout time and space. In doing this, she ensured that the Doctor and Rose's past selves would eventually follow the clues that would lead them to this moment, creating an ontological paradox. The Bad Wolf then destroyed the Dalek fleet and all the Daleks on Satellite Five, including the Dalek Emperor. She also altered reality to bring Captain Jack Harkness back from death. (TV: The Parting of the Ways)
Jack later discovered he could apparently no longer die by any means, and it took him nearly a century and a half of subjective time before he finally learned that she was responsible. (TV: Utopia) As the instigator of Jack's immortality, Bad Wolf was therefore indirectly responsible for the event known as Miracle Day on Earth. (TV: The New World, et al.)
Rose was unable to hold the power permanently without burning up, and the Ninth Doctor restored Rose's human self by absorbing the Vortex energy into his own body, a process that would prove destructive to his cellular structure and so prompt his next regeneration. (TV: The Parting of the Ways) Rose subsequently lost most of her memory of her time as the Bad Wolf, saying that the knowledge had been locked, as if it was "forbidden", and it was stated that Rose trying to repeat it could cause the universe to explode. (TV: The Christmas Invasion)
At some point during her travels with the Tenth Doctor, Rose learned of what she did as the Bad Wolf. Facing a lone Dalek at Crediton Vale, Rose recalled this only for the Doctor to deny her implicit suggestion, warning that another attempt could destroy the universe. (PROSE: I Am a Dalek) She later boasted that she killed the Dalek Emperor when she thought the Cult of Skaro were going to kill her soon anyway. (TV: Doomsday)
The Bad Wolf entity was at least a part of the Moment. She told this to Gabby Gonzalez, a companion of the Tenth Doctor, when she found her floating through the Vortex. Finding Gabby at a "crucial point" in the Tenth Doctor's life, she showed her the Doctor's past companions, and how they were all connected before depositing her into the Twelfth Doctor's TARDIS. (COMIC: The Good Companion)
Alternate timelines
In a number of alternate timelines visited by Siblings Same and Different whilst attempting to build the Paradox TARDIS, the Bad Wolf and the Daleks fused together, and spread across all of time and space as an army of metal copies of Rose Tyler. (PROSE: The Paradox Moon)
Abilities
The Bad Wolf was both nigh-omnipotent and omniscient. It had near-infinite manipulative abilities, including reality warping and fate manipulation, and as well as being immortal with immunity to anything which could negate it, it could also grant immortality to other lifeforms, dead or alive. (TV: The Parting of the Ways, Utopia) The Bad Wolf was also capable of extreme feats of destruction, such as reducing a Dalek Fleet comprised of five-hundred-thousand Bronze Daleks and two-hundred Dalek flying saucers - including the Dalek Emperor of the Last Great Time War and its own flagship - to mere atoms. It could also perceive the entirety of time and space, and spread its own name across various points in time and space so that Rose Tyler and the Ninth Doctor would encounter the words which would eventually lead Rose to becoming the Bad Wolf in the first place.
However, as the Bad Wolf was created from Rose Tyler, a human, the entity had the potential to lose control of its own power, (TV: The Parting of the Ways) which could lead to catastrophic consequences; indeed, in numerous alternate timelines, the Bad Wolf's inability to control its own powers led to it merging with the Dalek Emperor's Dalek army and infinitely replicating across all of time and space. (PROSE: The Paradox Moon) The Ninth Doctor remarked that no lifeform was capable of containing the full force of the Time Vortex, and speculated that a Time Lord with the Bad Wolf's abilities would become "a vengeful god". He absorbed the Time Vortex from the Bad Wolf in order to save Rose's life; even then, the Doctor himself was forced to regenerate to heal from the extreme cellular destruction that he experienced from absorbing such power. (TV: The Parting of the Ways)
Later references
Upon Rose Tyler's meeting with the Lupine Wavelength Haemovariform, the latter noted that she also had "something of the wolf" in her. (TV: Tooth and Claw)
After being captured by the Torchwood Institute, the Tenth Doctor pretended that Jackie Tyler was Rose, and claimed that she had stared into the heart of the Time Vortex and aged fifty-seven years. She replied with "I'm forty!". (TV: Army of Ghosts)
While on the Valiant, the Saxon Master commented to the Tenth Doctor how he "once had companions who could absorb the Time Vortex". (TV: Last of the Time Lords)
When Rose and the Tenth Doctor said good-bye to each other after the former became stranded in "Pete's World", they met on a beach in said parallel universe's Earth in Norway called Dårlig Ulv Stranden, which translated in English as "Bad Wolf Bay". (TV: Doomsday) She said goodbye to the Tenth Doctor there a second time after helping him and numerous other companions of the Doctor save the multiverse from the Reality Bomb. (TV: Journey's End)
In "Donna's World", Rose whispered the words "Bad Wolf" to a dying Donna Noble. After Donna remembered these words in N-Space, all the banners and signs in the Shan Shen marketplace read "BAD WOLF", including the sign on the Doctor's TARDIS. (TV: Turn Left) This warned the Tenth Doctor and Donna of the impending danger of Davros and the New Dalek Empire. (TV: The Stolen Earth)
All of the Torchwood Institute's records on file of the Doctor's companions were reportedly destroyed or corrupted in some way by a virus known as the "Bad Wolf virus". (TV: Love & Monsters)
When the consciousness of the Moment took form, it selected the form of Rose Tyler as the Bad Wolf entity. When the War Doctor commented that he could kiss her for giving him a way out of the dual-genocide he was prepared to commit, she remarked, "That's definitely gonna happen." The Tenth Doctor was visibly surprised to hear his past self mention the Bad Wolf, as he was unaware of the presence of Bad Wolf's likeness, and also because the War Doctor (as far as the Tenth and Eleventh Doctors remembered) had never encountered Bad Wolf, nor Rose Tyler. Due to the timelines rearranging themselves, the War Doctor (and his immediate successor) retained no memory of his meeting "Bad Wolf". (TV: The Day of the Doctor)