My “inuniverse” Rationalization of how Son Goku was able to harness and absorb the Godlike Ki in Battle of Gods and then Transform without the ritual:
In Battle of Gods, the first time Goku was transformed through the help of his friends and family. Thus, his body and spirit felt and learned the god like ki. The feeling or memories of that godlike ki were akin to his dormant memories of his power when he turned into an Oozaru that he had when he drank the Super God Water during the Piccolo Daimao Arc. According to the Super Exciting Guide the memories of that power released it's dormancy or at the very least the memories associated power with the oozaru form. Either way memory (or ki/physical memory) is what contributed to Goku's boost back then, and I rationalize is the same association that attributed to his absorbiton of the God Like power in BOG or at least allowing him to be able to harness that power despite the transformation itself timing out.
Consequently, while battling Beerus, although the form timed out, Goku was able to retain much of that power because he had the feeling of that Ki within him and had absorbed the power. This could also explain why, Goku in the finale, is able to reappear in the form with red hair and exciting aura: he had the memories and the ki was triggered by memories of the ritual and encouragement of his friends as that is what Toriyama has said Ki can come from.
I base my reasoning simply on the manga itself and the Super Exciting guide's 2 explicit postulates:
Proposition 1: Memory of Power can be used to channel power and achieve higher status:
Herms wrote:The entry on the Super God Water explains why that image of an Oozaru appeared when Goku’s power awakened: it says that the poison triggered a memory within Goku associating the tremendous power sleeping inside him with the Oozaru (or possibly just that it triggered a memory sleeping inside him associating tremendous power with the Oozaru; I’m not too sure which it means).
Proposition 2: Ki enhancement can come from courage and others encouragement:
Herms wrote:What’s the secret of winning in battle?
When it comes to battle, the most important thing is ki size, and its control. Of course, “ki” also includes such spiritual power as energy/vigor [genki] and bravery [yuuki], and being in one’s right mind [shouki; could also be translated as "true character"] (note 11). There’s a limit to physical strength, no matter how much you toughen it up, and the only way to overcome that it is with “ki”. I think that it was through turning ki into formidable power that Goku drew closer to being the strongest warrior in the universe.
Note 11: Genki and yuuki, shouki
The word “energy” [genki; also “health”, “vigor”, etc] is well known, but if you had an illness [byouki], would you not have energy? That’s difficult to answer. There are some people who have a healthy body but no energy, but there are also some people who are full of energy even when ill. What’s more, even with people who have about the same amount of power, there are some who, when attacked by Piccolo Daimao for instance, will stand firm and tough it out, while others will run away trembling in fear. If you think this way, you might understand how different kinds of “ki” are very important in battle. In the world of Chinese kenpo disciplines such as Tai Chi, much importance is placed on the training and control of ki.
Although in “DragonBall” the wizard Babidi had invented machines that measured and extracted “ki”, modern science is unable to measure “ki” or store it anywhere. However, there are times when people support you, and give you energy [genki] and courage [yuuki]. Though it can’t be seen, what is present then is “ki”.
