You have enabled the Spotlight module within your Aurea Social environment and want to use the Personal Badges Stream filter as the default filter within your Activity Stream web part. When you review the Newsfeed, you see the Personal Badge filter as expected, but it does not appear as a "default filter" option within the Web Part properties.
While enabling the Spotlight module allows the Personal Badges stream filter, it is not one of the default options available within the Activity Stream web part. There is no way to configure the web part to reference the Personal Badges filter as the default.
Note: If you do not see the Personal Badged filter within the Stream filters, please see User Badges Not Displaying in Newsfeed Filter Options to investigate enabling the Spotlight feature.