In traditional research labs, all animal research requires IRB approval. The IRB certifies that, among other things, the animals are appropriately housed, the research is not excessively cruel, and that the suffering of the animals is justified given the nature of the research. For you it would be very difficult to receive IRB approval because, for instance, you must have a certified facility for keeping the animals that meets government regulations, which is prohibitively expensive to do for a private person. Without IRB approval, no journal will consider your research for publication, and you cannot go through any government process to, for instance, receive a patent for a treatment or license a drug.
If you don't care about any of this, and are motivated only by personal curiosity, in principle I don't think there is much stopping you from simply feeding some harmless medicines or food additives to your pet rats. For instance, people "experiment" on their pets all the time, such as trying different diets (like raw diets for dogs) and many pet medicines do not require prescriptions or permission from a vet. Of course if your rats suffer as a result of the testing, or if you knowingly administer drugs with harmful side effects, you may run afoul animal cruelty laws as they apply to pet ownership. I would still caution against this, I think it is unethical to endanger the well being of your pet for any reason.
However, I doubt your research will succeed. Testing on rats and mice is already difficult and costly, lifespan research is even harder due to very long period of experimentation. You also typically need a large number of animals to get meaningful results, due to substantial individual variation, even in very standardized lab conditions. Moreover, rats and mice are very different from humans, especially with regard to lifespan, so the applicability of your research to humans would be very dubious. You might be able to obtain some results that seem like they prove a hypothesis, but in reality they would be artifacts, because you have no way of adequately controlling for many confounds inherent in such work. Showing anything conclusively with mice and rats is a challenge even for top research labs with tremendous resources and experience.
And here comes the disclaimer: I am not a lawyer and this post is not legal advice. The information is provided for entertainment purposes only. Please do not go and start a rat experiment in your backyard because of what I said. Knowingly causing harm or suffering to your pets is unethical and immoral, and probably illegal as well. If you are interested in doing animal research, I would suggest you to get in touch with a lawyer or researcher with experience in this area.