A well-written letter and other methods of persuasion can help reduce the cost of expensive hits and produce unforgettable results.Beau, the addled midlife wreck played by Joaquin Phoenix in “Beau Is Afraid,” isn’t just afraid, he is terrorized: harassed, beaten, stabbed and even kidnapped in a surreal black comedy that often feels less like a conventional film than a three-hour panic attack. (In the hands of high-anxiety auteur Ari Aster, of “Hereditary” and “Midsommar” fame, consider that a compliment.)Thanks to his monstrous mother, he has become a man resigned to life without love or companionship. Then, deep in the movie comes a reprieve — a late chance at romance with his childhood crush (Parker Posey), soundtracked, incongruously, to the lilting strains of Mariah Carey’s smash 1995 ballad “Always Be My Baby.” Things go obscenely, catastrophically awry from there, as they are wont to do with Beau, but the song plays on.When the scene played at a preview at the Metrograph Theater in Manhattan this month, a packed room of industry insiders, press and celebrities that included Phoenix and the actor Robert Pattinson collectively gasped in recognition, then cheered. How exactly did the queen of five-octave pop end up here? For Aster, it turns out, there was never a second choice.“Ari had written a first draft of the script over 10 years ago, and ‘Always Be My Baby’ was in it from the very beginning,” said his production partner, Lars Knudsen, who also works frequently with filmmakers like Robert Eggers (“The Northman”) and Mike Mills (“Beginners”). “I honestly didn’t know how integral and crucial it was to him to have that song until we were in the edit, but we knew that it was going be very expensive, and that Mariah might not approve it. There was a feeling like, ‘Look, we’ll try, but we likely won’t be able to afford it.’”Nevertheless, Aster penned what Knudsen called “a very beautifully written letter” to the singer and pleaded his case; improbably, she said yes. When she first received the request, Carey said via email, “I was quite intrigued. Then, as I watched the scene, I was a bit shocked at first because of my prudish nature (ha!), but immediately understood the importance of that particular moment.”The writer-director Ari Aster wrote a letter to Carey, one strategy indie directors use when they can’t afford a song.A24She continued, “I’m really happy with the way people are responding to it, and thrilled that Ari is being recognized for his talent, creativity and artistic vision.”(Several days after the Metrograph screening, Carey briefly lit the internet ablaze when she appeared beaming on the red carpet alongside Posey and Aster at the film’s official New York premiere, resplendent in black leather.)“Beau” is perhaps the most prominent recent example of indie movies — many of which seem to stem from the tastemaking studio A24 — that stake their wildest hopes on finagling the rights to an instantly recognizable and often formidably expensive pop song. When the pairing goes well, it can be a zeitgeist-y boon for the kind of projects that rely more on word of mouth than marketing (in addition, of course, to fulfilling the highly specific vision of their creators).Think of ’N Sync’s elastic boy-band anthem “Bye Bye Bye,” which runs prominently throughout “Red Rocket,” the 2021 festival hit from writer-director Sean Baker (“The Florida Project”) about a washed-up porn star, or Paris Hilton’s featherweight bop “Stars Are Blind,” which provides a rare moment of levity in the bleak hard-candy noir of Emerald Fennell’s 2020 “Promising Young Woman.”A movie like “Guardians of the Galaxy” has Marvel Studios to underwrite its wall-to-wall usage of hits by David Bowie, the Jackson 5 and Marvin Gaye, among others. (The franchise’s director, James Gunn, once said he had paid “a million dollars” for a single song.) For small independent projects like “Aftersun,” though, the dreamy, elliptical father-daughter drama by the first-time director Charlotte Wells, a track like Queen and Bowie’s anthem “Under Pressure” — used to harrowing effect in a climactic scene — can easily consume the entire budget.That’s where highly personal appeals to the artist or estate with rights to the song — and no small amount of serendipity — often come into play. For “American Honey” (2016), a sexually frank verité road trip with a largely unknown cast, the British director Andrea Arnold had little choice but to get permission after the fact, or recut the film entirely; tracks including Rihanna and Calvin Harris’s “We Found Love” were not overlaid but woven into scenes that had already been shot.In that case, said Knudsen, who also produced “Honey,” both artists were moved enough by the material to not only give their permission, but also provide a sort of friends-and-family discount: “If it had been made by a bigger studio, then obviously we would have to pay” full price, he said. “But because we were an under-five-million-dollar movie with a reputable director who was trying to tell this very personal story where that song was the center of it, I think it definitely helped.”In the right circumstances, of course, less-expected collaborations like these can very much serve the musicians, too, even when they reduce their fees — a feedback loop of indie cred and mainstream appeal that confers fresh relevance to both parties.“When you make a convincing case, the publishing companies and the artists do understand,” Knudsen said. “I mean, ‘American Honey’ played in competition at Cannes, and A24 released it. If there wasn’t a sliding scale, then no independent film would be able to have any of these songs in their movies.”For directors like Arnold or Aster, those scenes become signatures. And for a certain kind of cinephile, “those songs will just have a very different place in their hearts. So that’s good for everyone, right?”Little Indie, Big Song“Aftersun”: David Bowie and Queen’s classic “Under Pressure” underpins the emotional climax of this impressionistic 2022 drama, which earned Paul Mescal an Oscar nomination for best actor.“Red Rocket”: ’N Sync’s 2000 hit “Bye Bye Bye” bookends this scrappy 2021 film, a character study of a prodigal porn star (Simon Rex) returning to his Texas roots.“Promising Young Woman”: Paris Hilton’s “Stars Are Blind” provides a rare moment of connection for two damaged characters in this highly stylized 2020 neofeminist revenge tale.“American Honey”: The Rihanna-Calvin Harris banger “We Found Love” becomes a sort of central theme for conflicted lovers played by played by Sasha Lane and Shia LaBeouf in this 2016 road movie.“Spring Breakers”: Britney Spears’s lachrymose 2003 ballad “Everytime” plays as a girl gang in pink balaclavas goes on a crime spree led by a demented James Franco in this 2013 nihilist comedy. More