Domain Age
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Hello I try to learn seo and find out this kind of problem. I have new domain it is registered 3 month ago and have different domain which register 3 years ago. One website is indexed and new website is not indexed. New domain have more backlinks and ahrefs authority. Doe domain age and domain expire time give any points to fast indexing and ranking ?
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In my experience, yes. Domain age is a factor, especially when it has backlinks pointing to it. Just make sure they are high quality, relevant websites and the domain isn't penalized in any way.
That being said, you can index a new site almost just as quickly if you know what to do (ping google, share on social media, bookmark).
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I would disagree to an extent. Whist it may be a factor if you go buy an old domain and build new content on it Google pretty much treats it as a new domain and along with that if all the links going to that domain are for example about cats and your domain is about dogs you're making your life harder.
Matt Cutts mentioned here that its not something you should worry about, now while he neither confirms it, nor denies it what it does indicate are there are other areas to focus on.
So in short if the domain is the same niche and the links already built are a benefit the the user (who may get to your site from those links) then it should be fine but at the same time is it worth the risk of moving a whole site? There are many good ways of indexing a site which is a lot easier than a redirect.
Work on the content of your site, get it shared (even socially) and you will find it ranks quicker (also try 'Fetch as Google') than just redirecting a domain to it (or visa versa)
Hope that helps.