Simon Cowell & Lauren Silverman

Simon Cowell knows more about caring for infants than you think, and says ... medically, there's NO issue with his little beach vaca with his baby.

Simon and Lauren Silverman were out for dinner last night in Miami -- sans baby Eric -- when he fired back at critics (like us) who said it probably wasn't a great idea to take their 10-day-old baby out for a day on the beach.

Simon says Eric is "very healthy" ... and added the whole trip was doc approved.

Simon Cowell's two dogs are puppies non grata on the beach in Miami -- the TV mogul was just warned by cops to get them off the sand ... or else.

Simon's been chilling in the sun all day with GF Lauren Silverman and their newborn son Eric -- but despite the beach's explicit NO DOGS ALLOWED rule, he brought along his two Yorkshire Terriers Squiddly and Diddly.

We're told a police officer gave Simon an official warning ... and Simon and Lauren took off soon after.

We almost felt bad for the dogs ... almost.

Simon Cowell was sent packing Friday ... just days after his baby was born, but not because Lauren Silverman threw him out -- because a judge laid down the law in her complicated custody case.

As we previously reported ... Lauren's nasty divorce from Andrew Silverman ended with a custody arrangement for their 7-year-old son Adam, which required Lauren to keep Simon away from the kid for a year. Adam stayed with Lauren over the last few days, which meant Simon needed to skedaddle.

Simon moved into the swanky St. Regis Hotel in NYC ... but we're told he moved back with Lauren when Adam left.

As for the immediate future ... we're told Simon will stay in NYC for 4 to 5 weeks, until their baby is allowed to fly. The new family will then go to London for "X Factor" auditions. They'll be in the UK for around 3 weeks and then return to NYC.

Our Simon sources say he and Lauren will split their time between NYC and London, and since he has a plane it's no big deal.

Simon Cowell flashed his new daddy skills -- in front of all the paparazzi -- and FAILED MISERABLY at the dreaded car seat test.

Simon and Lauren Silverman took their newborn Eric to the doctor Friday for a checkup. Lauren said the baby's doing great -- and then it was Simon's time to shine ... or not.

Watch what happens when he attempts to load the kid into the car. If we're being honest ... Papa Cowell made a big no-no.

Simon Cowell's new son has been cleared for takeoff -- TMZ has learned, the 5-day-old child just got his very first passport ... to become one of the most well-traveled jetsetting babies in the world.

Simon and Lauren just left an NYC courthouse, where they filed papers to get their son Eric his passport.

We're told BOTH parents had to be present in order to get the official government paperwork -- and since Simon's only in town for a few days, the couple decided to check this off their to-do list right away.

Makes sense -- Simon spends tons of time in the UK ... so baby Eric will obviously be making plenty of trips across the pond.

Simon Cowell has a male in his life that is actually showing more chest than him -- and his name is Eric.

It's a pretty shocking sight for a man who by all accounts has avoided parenthood for 54 years.

Simon boasted how handsome his son is, adding, "I never knew how much love and pride I would feel."

Simon, Lauren Silverman and 2-day-old Eric posed for the pic in the NYC hospital where the kid was born.

So how do you feel today ... looking at a 2-day-old kid who will never have to work a day in his life?

4:25 PM PT -- the baby's name is ERIC ... after Simon's father.

Simon Cowell is officially a father ... his pregnant GF Lauren Silverman has finally popped in NYC.

36-year-old Silverman just gave birth to a healthy baby boy at Lenox Hill Hospital -- after Simon flew across the ocean from London to be by her side for the delivery. As we reported, Lauren went into labor early this morning and Simon hopped on a plane as soon as he heard.

We're told the baby was born at 5:45 PM EST at 6 pounds, 7 ounces.

The boy is 54-year-old Simon's first child and Lauren's second -- she also has an 8-year-old son with her ex Andrew Silverman.

So far, no word on the baby's name.

Simon Cowell's baby mama better tighten up her lady parts ... 'cause the "X-Factor" judge just landed in New York -- and he's making a beeline for the hospital.

We broke the story ... Lauren Silverman went into labor early Friday morning -- and Simon rushed home from London to be there -- but based on his look in the pic ... the soon to be dad doesn't look too worried about missing things.

Something tells us he won't be filming the gruesome process.

7:25 AM PT -- Daddy's gonna make it ... our maternity sources tell us Simon is zipping across the pond right now, and Lauren is early enough in labor that docs will delay the birth until he arrives.

Simon Cowell's first baby is on the way -- his girlfriend Lauren Silverman is in labor, and has checked into a New York City hospital ... and Simon is racing to get there before the kid arrives.

Sources close to Lauren tell TMZ she went to Lenox Hill hospital in Manhattan (Upper East Side, of course) within the last few hours -- and immediately sent up the baby signal ... alerting Simon to get on a private jet bound for NYC.

He's been in London shooting one of his competition TV shows.

If Lauren has the baby today -- they're expecting a boy -- we're told she'd be about a week and a half early, but it's nothing to worry about.

Happy Valentine's Day, Simon.

Story developing ...

Simon Cowell's baby mama, Lauren Silverman, signed a divorce settlement agreement which has a shocking penalty -- if Simon goes near her 7-year-old son, she has to pay her ex a $50,000 PENALTY!

TMZ is privy to the unbelievable settlement agreement, which also provides that if Lauren violates the Simon stay-away clause 3 times, she loses all contact with her son, Adam, until a court deals with the violations.

The Simon stay-away clause only lasts until January, 2015. So presumably the kid can be around Simon in a year ... if he's still part of Lauren's life.

But the plot thickens. The settlement agreement also provides Andrew Silverman, Lauren's ex, cannot trash Simon to Adam and has a legal obligation to make sure no one else around him talks smack.

And there's a clause in the settlement that provides Lauren and Andrew both agree to prevent Adam from calling anyone other than them "mother" or "father." Translation: Simon is not your daddy.

Andrew agrees to pay Lauren $3k a month in child support ... a modest amount and it goes to what TMZ has been reporting ... that Simon will become the cash cow.

Finally, Lauren and Andrew agreed to change the cause of divorce from adultery to "broken down irretrievably."

Simon Cowell's baby mama Lauren Silverman and her ex won't be settling any future disputes through overpaid lawyers -- they've enlisted a trusted mutual friend to hash out their differences ... a man who also happens to be a famous BILLIONAIRE.

According to the divorce settlement, Lauren and Andrew Silverman have named Sir Philip Green as a mediator. Green is the current CEO of the Arcadia Group, which owns the likes of Topshop, and is worth a reported $5.3 billion.

Under the settlement, Green will be one of two mediators (the other being attorney Martin Edelman) for the next 5 years.

We're told the reason Green was named a mediator is because Lauren, Andrew, and Simon each consider him to be a close friend -- and feel comfortable using Green as a neutral go-between.

Two interesting things about this video with Simon Cowell and the woman carrying his fetus, Lauren Silverman. And it's all about who is responsible for what in their relationship.

Lauren's responsibilities -- according to the couple, Lauren is tasked with making the decision about filming the birth of their child.

*Note -- Our camera guy has volunteered for the job ... and Lauren didn't exactly turn him down.

Simon's responsibilities -- Lauren says the decision to get married is entirely up to him.

Duh.

Simon Cowell might be over the shock of becoming a father, at 53, but his reaction to possibly getting married on top of that is down right hysterical -- as in ... "HAHAHAHAHAHAHA!!!"

Simon called in to "TMZ Live" to discuss a possible marriage to baby mama Lauren Silverman -- and his feelings about creating his own little Mini-Me.

Simon boasted ... "If it's a boy, then there will be another me on this earth for many, many more years to come."

As for his reaction to the news he'd knocked up Lauren ... "Gulp!"

But you gotta see Simon's response to the marriage question. Classic ... bachelor.

Simon Cowell was being smothered by his pregnant baby mama Lauren Silverman in public during a beach trip in St. Tropez this week ... which means one of two things.

1) Simon has finally come to terms with fatherhood and wants to show everyone how much he cares about Lauren.

2) He's totally passed out

Simon Cowell, Lauren Silverman and their unborn spawn are making it official ... stepping out together for the first time IN PUBLIC since news of their relationship broke.

The pic was snapped in the South of France Saturday -- where the two are vacationing together amidst other rich people ... looking carefree and happy.