Around the time my second baby was born, in 2014, I started seeing ads for a new air-permeable crib mattress. It featured a baby sleeping face down with reassurance that she could safely breathe right through the mattress. With a gesture to sudden infant death syndrome and suffocation, it urged me to “avoid the possible threat of waking up to a tragedy.”

Like many parents with new babies, I have a great fear of SIDS. One of the best ways to reduce the risk of the still-mysterious disease is to put babies down to sleep on their backs, but between about 4 and 6 months, babies learn to roll onto their bellies. The American Academy of Pediatrics reassures us that as long as they’re being put down on their backs in a safe sleep environment, they’re fine, particularly as more than 70 percent of SIDS and suffocation deaths occur before 4 months of age.

Still, the thought that a mattress could help further reduce the risk was appealing. The mattress that served us well for our first baby cost $80, met all government safety standards, and was still in good shape. But now I wondered if its vinyl cover—so easy to clean after a middle-of-the-night poop explosion—was a danger to my new baby. Should we invest in a breathable mattress? (And if so, what should we do with our old crib mattress—donate it to a less-fortunate baby whose family couldn’t afford the cost of a breathable one?)

There was no shortage of options. Several companies now sell models of “breathable” crib mattresses, all offering the peace of mind of safer sleep for the price of about $300–$400. One company, Secure Beginnings, claims that babies are in danger when they start rolling onto their bellies and that its mattress mitigates that risk. “Rolling over should not be a death sentence!” it wrote on their Facebook page.

It should not be. But does the science back these mattress sellers’ claims up? The advertising sure wants you to think so, but a closer look at those claims reveals troubling holes.

Secure Beginnings thinks that the main problem with babies rolling onto their tummies is increased risk of rebreathing, which is what happens when exhaled carbon dioxide builds up in the immediate environment around a baby’s face. It and another company, Newton, both commissioned third-party labs to test this using a baby-sized mannequin “breathing” facedown into their mattresses. The data showed that “exhaled” carbon dioxide disperses more quickly on their breathable mattresses than on traditional crib mattresses.

The question is if this mannequin measurement is of consequence to a real baby. Nobody can answer this for sure, because we aren’t certain what happens when a baby dies of SIDS. There was a lot of research in the 1990s that showed rebreathing is a hazard when babies sleep on soft bedding, like sheepskin, or with a blanket covering their heads. However, there isn’t evidence that sleeping on firm mattresses with a tight-fitting sheets—as is recommended—causes problematic rebreathing in babies, even if they roll onto their tummies. Plus, rebreathing is just one hypothesis for why tummy sleeping might be riskier. More recent research has found that babies sleeping belly-down also have impaired arousability, altered cardiovascular control, and lower blood pressure and cerebral oxygenation, any of which may be important when it comes to SIDS. “No mattress can ameliorate these alterations in infant physiology,” wrote Rosemary Horne, a researcher at Monash University in Australia who has led many of the studies on prone sleep, in an email.

One problem with mattresses being marketed as “breathable” is that there aren’t any approved standards for what this should mean. Another company, Nook, says it designs its crib mattresses “with one goal in mind: to reduce your baby’s SIDS risk as much as possible.” To that end, Nook fills the Pebble Pure mattress ($395) with coconut husk. Dr. James Kemp, a pediatric pulmonologist at St. Louis Children’s Hospital, called this “exactly the wrong idea,” in an email, noting that a porous interior might actually trap carbon dioxide. A mattress filled with tea tree bark made in Australia in the 1980s and ’90s and marketed as breathable turned out to increase the risk of SIDS, and, according to a study by Kemp, allowed more rebreathing. (Nook did not respond to my request for test results on its mattress nor my request for comment by press time.)

Newton claims its breathable crib mattress also eliminates the risk of overheating, which has been associated with SIDS. The company couldn’t provide data on this, but a spokeswoman said that it was a “basic scientific principle.” She referred me to a video of a rice-filled sock heated to 130 degrees Fahrenheit and placed on their mattress to see how long it took to cool to 98 F. That’s neat, but my baby isn’t a sock, and if he were 130 F I would have other problems. Instead, I looked at a 2007 study in which U.K. researchers measured the body temperature of real sleeping babies. They found that babies were actually warmer on air-permeable mattresses compared with impermeable ones. That seems counterintuitive, but that’s why we need hard data: Our intuition is often wrong.

In designing “breathable” mattresses and claiming that they will inherently keep babies safer, these companies are demonstrating a fundamental lack of understanding of the complexity of SIDS, and, more broadly, of science. “The ‘evidence’ the manufacturers of these crib mattresses provide is not evidence of reduced risk of suffocation or SIDS,” said Dr. Fern Hauck, a member of the AAP’s Task Force on SIDS. “As long as the mattresses meet government safety standards, they can be used, but they should not be marketed as preventing SIDS or suffocation,” she added.

The Food and Drug Administration agrees. If a company claims that its product reduces the risk of SIDS or suffocation, then it is, by definition, a medical device and needs FDA approval. That requires evidence it actually works as claimed, and so far, for these products, that evidence is lacking. The FDA has never approved a product to reduce the risk of SIDS, so a company that makes this claim could be in violation of the Federal Food, Drug, and Cosmetic Act. This includes claims that are direct or implied, and if an ad mentions SIDS and a safer mattress, that implication sends a message to anxious parents.

One scary concern SIDS researchers brought up was the worry that parents using these mattresses might think the products are so effective that it is OK to put their babies down on their bellies. The companies don’t directly promote this idea, but by touting breathability as a safety feature, you can imagine how a sleep-deprived parent might get the wrong takeaway.

When I talked to Julie Andreae, president of Secure Beginnings, she dismissed this concern. Yet the company’s own website not only shows photos of babies sleeping facedown; it also includes the following testimonial:

When my daughter was born 4 months ago, I was too fearful of SIDS to place her on her belly to sleep on a regular mattress, and was beyond delighted to discover this breathable crib mattress! We use it exclusively, and I sleep more soundly knowing my baby is sleeping safely on her belly, her side, and on her back.

But regardless of what the parent may think, the breathable mattress didn’t make her baby safer.*

Kemp advises parents to focus on SIDS prevention strategies that are evidence-based, and incidentally, won’t blow your baby budget: Put your baby down for sleep on her back; in her own sleep space; and on a firm mattress free of loose bedding, bumper pads, or other objects. Beyond that, buying a fancy mattress is not about keeping the baby safe but rather about soothing parents’ fears. “Much of what we do for our kids is really to reduce our own anxiety,” Kemp told me.

And we have plenty of that on our own, without companies trying to add to it.

*Update, Aug. 25, 2016: This piece has been updated to clarify that the purchase of a breathable mattress did not make the customer’s baby safer.

In addition to the sources mentioned, the author would also like to thank Dr. Benjamin Hoffman, pediatrician at Doernbecher Children’s Hospital at Oregon Health and Science University; Peter Fleming, professor of infant health and developmental physiology at the University of Bristol in the United Kingdom; and Peter Blair, professor of medical statistics at the University of Bristol.