How do IDP recognize a previously authenticated user?

Dear all,
sorry for my English is not good.

In order to implement SSO with SourceID.Java, I imagine this situation:
a person wants to use some services supplied by SP1 and SP2.
When the browser makes an http request to SP1, SP1 has to verify
user authentication. In order to authenticate the user, it redirects the
browser to IDP.
Now user can fill its credentials, for example, to IDP and IDP redirects
the user to SP1.
User authentication is recognized by SP1.
If user wants to make different http requests in a short period of time,
how
can SP1 "know" user http session state? I suppose SP1 sends cookies to
the browser.
Now I imagine the same person to contact SP2 for other services. How do
SP2 know  that the user was already authenticated in the same circle of
trust?
I suppose SP2 redirect the browser to IDP for authentication. But now
how can
IDP recognise that the browser was previously used to authenticate the
same user?
Did IDP send a cookie to the browser in the response to previous request?

I thank you in advance and sorry if my questions are already done.

Antonio