I saw messages that you can change the build on Heroku, use PhantomJS with the Cedar stack, but can I launch a mute browser on the unicorn stack?
Yes, and someone already wrote buildpack to enable PhantomJS (the Internet and open source are amazing).
If you want to run the headless PhantomJS on the Heroku dynamo, you will need to:
A multi-line package should be configured as an official Heroku buildpack. Then you need to add two other buildpack urls to your .buildpacks file. There is additional information about configuring each collector using the links above.
.buildpacks
Using this, you can deploy the Rails application, run your PhantomJS code / tests, all on one headless announcer.