Hi there!
Sam from Moz's Help Team here!
So, newly discovered links have the ability to be populated into our index in about 1-3 days, however there are a lot of factors which can affect our ability to find and index links to your site. It's important to note that we add new data to our index everyday but it may take some time for us to discover backlinks to your site based on factors like crawlability of the referring pages, quality of the links and the referring pages, and more.
If you are not seeing links that you know you have, you may want to make sure that they can be indexed. It is also a good idea to check to see if we've indexed the page on which that link is found. If we haven't indexed the referring page yet, you won't see your link in our index
You can also add links to Link Tracking Lists. Once you add a link to your tracking lists we will add that page to be crawled. As long as it is accessible to our crawler, you should see the link in our index as soon as we can index those pages.
Lastly, I have a great guide here with some things to check around why we may not have found your links yet: https://moz.com/help/link-explorer/link-building/moz-isnt-finding-your-links
In terms of Domain Authority - Domain Authority in Link Explorer and the Links section of your Moz Campaign correlates with Google rankings to give you an accurate representation of a sites's ranking power.
To understand how to improve DA (Domain Authority) it can help to get some insight into how we gather our data, which I'll include below. Sorry in advance for the long message!
To calculate DA, we index the web by following links using a crawler. Our crawler is built on a machine-learning based model that is optimized to select pages like those that appear in our collection of Google SERPs. We feed the machine learning model with features of the URL like the backlink counts for the URL and the PLD (pay-level domains), features about the URL like its length and how many subdirectories it has, and features on the quality of the domains linking to the URL and PLD. So, it's not based on any one particular metric, but we're training the crawler to start with high-value links.
In terms of improving your Domain Authority, with the information above in mind, it's best to look at both you on-page SEO to make sure you've covered the basics.
Then you also want to look at your off-site SEO and building more links more links to your site.
There are lots of different areas of Link Building to explore, from patching up broken links, improving your internal link profile, and good ol' fashioned content creation.
Here is a great video by our founder Rand which goes over some easy link building tactics to get you started.
I hope that helps to start you off! Feel free to reach out to help@moz.com if you have any other questions!