We need to default URL to unique name. If it is www then with no prefix or vice versa. So decision to be made is either stick with www or with no prefix.
With no prefix cookie is set for all sub domains. What are other downsides for it? Or benefits?
Basically we need this for OpenID as OpenID will make users look different if they came from www or with no prefix.
As our site is new so we can go with either one. Also, how the domain name looks is not much of a concern.
You probably want to redirect (with a HTTP 301 - Permanent Redirect) one to the other anyway, since maintaining consistent urls is much easier that way. So whichever you decide, just make sure the actual authentication is done after the redirect, and users looking different won't be an issue.
That said, if you want www or not depends entirely on how other things in your appliction works. You mention that cookies for domain.com will be saved for all subdomains - is this something you want? Are you ever going to need to differentiate (for example, by allowing users to set up their own authentication systems for subdomains as a shared hosting service might do)?
If none of the differences you find between including and excluding www matter to your application, I'd go for not using www. The main reason for this is my picture of current trends on the internet - more and more applications (SO is an example of this) tend to leave the www out, both when linking to their own sites, and in marketing of different kinds.
However, the main point is make both work. You don't want your site to break because the user did(n't) type www at the beginning of the url.
By not using the www subdomain, you can suffer a performance hit when delivering static content, as noted here: http://developer.yahoo.com/performance/rules.html#cookie_free. As I understand it, if you use http://example.com/ and http://static.example.com for static content, any cookies you set on the main domain will be passed with requests to your static subdomain.
This can be avoided quite easily, by buying a distinct domain for static content. However, this can certainly be dealt with by using a www subdomain.
Then again, this is a very minor con, and really only comes into play when you're dealing with a high-demand site. (For example, Digg uses http://digg.com and http://*.diggstatic.com).
Ultimately, I would say that this is such a minor problem that it can probably be dealt with if performance starts to suffer. Don't optimize prematurely, and all that...
And, as @Tomas Lycken points out, make sure you account for www even if you don't use the subdomain.