Gregg DeGuire/PictureGroup
Mike Fleeman
January 27, 2012 04:35 PM

As paramedics neared Demi Moore‘s front gate, one of the people on the line to 911 can be heard shouting out a name.

“Hey, Ru,” says the man. “What’s the name at the gate so that we can buzz them in from here?”

The dramatic emergency call for help last Monday night, after Moore went into convulsions when she “smoked something,” never reveals more about “Ru’s” identity, but this could be a reference to Moore’s oldest daughter Rumer Willis.

Rumer, 23, goes by that nickname and uses it on her Twitter account. And it’s Rumer who has most often been at her mother’s side in the months since the actress’s split from Ashton Kutcher, 33, after six years of marriage.

RELATED: Rumer Willis Spends Weekend Partying in L.A.

“Rumer’s spent more time with her mom in the last few months than she has in a while,” a source tells PEOPLE.

Rumer Willis, at the hospital
Splash News Online

In December, Rumer – whose dad is Moore’s ex, Bruce Willis – and her mom partied poolside at the Parrot Cay resort in the Turks and Caicos, and on Jan. 11 they shared a late night out together at L.A. hot spot Beacher’s Madhouse.

But it was a much different side of Rumer spotted in the days after Moore, 49, was taken away by ambulance and hospitalized. Appearing somber, she was seen outside an L.A.-area hospital earlier this week wearing a visitor’s sticker.

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