Democratic presidential candidate Barack Obama arrives in Indiana at 6:30 p.m. Tuesday night and he will remain in the state until 3:30 p.m. Wednesday.

The question of the day is whether he will leave with a running mate.

Currently, he has one event on his schedule — a town hall meeting in Elkhart, a city that has been hit hard by the auto-industry downturn, where he will be introduced by Indiana Senator Evan Bayh.

So what will Obama do with the rest of his day in Indiana?

He could, of course, sneak over to Chicago — only a couple hours drive or a few minutes flight from Elkhart — for more meetings with vice presidential prospects.

Or he could, as feverish media speculation suggests, simply announce that Bayh is his pick. The chatter went over the top last night, with various outlets — including CBS — breathlessly highlighting the fact that the Indiana senator’s staff softball team, the Bayh Partisans, had rescheduled a game.

Indianapolis Star writer Mary Beth Schneider is more measured in her assessment. Still, she noted Tuesday morning after reviewing the Obama campaign’s nearly wide-open schedule for Indiana that the presidential candidate will have “plenty of time, say, for a stop in Bayh’s hometown of Shirkieville in Vigo County or elsewhere to launch a ticket.”

The problem with the scenario is that, by most accounts, Obama is still weighing a number of prospects — and has yet to have sit-down meetings with most of them. Indeed, there is a good deal of talk at this point that — as he has taken more hits from McCain and suffered some poll fluctuation after his world tour — the presumptive Democratic nominee for president is now looking at a longer list of potential running mates.

That list is universally believed to include Delaware Senator Joe Biden, Virginia Governor Tim Kaine (last week’s favorite) and Bayh (this week’s favorite). But there is much talk, as well of Connecticut Senator Chris Dodd, former Georgia Senator Sam Nunn and Kansas Governor Kathleen Sebelius. And, of course, the name of New York Senator Hillary Clinton, who Bayh actually backed for the nomination, is somewhere in the mix.

Meanwhile, back in Indiana, neither the Obama campaign nor the Bayh camp has done much to tamp down speculation that Wednesday — which is just about the last day to make a veep pick before the Olympics begin a two-week run as the dominant story of the summer’s slowest month — will be the big day.

Bayh has done more than anyone to foster the speculation.

Indeed, the senator’s answers to question about whether he will be Barack Obama’s vice presidential running-mate sound like confirmations.

“Well, that’s the kind of thing you do say ‘yes’ to,” the Indiana senator chirped in an interview with Fox News, a network that’s unusually friendly to Bayh, a centrist whose willingness to side with Republican in the Senate on key tests regarding Iraq, civil liberties and trade policy has tended to endear him to conservatives.

Asked whether Obama’s aides are vetting him as a veep prospect, Bayh teased: “I’d love to answer your question, but I think I really can’t.”

That was in line with what Bayh has been saying in recent days.

Famously, he declared while campaigning with Obama in July that, “Any questions about the vice presidential thing are understandable and it’s good for my ego. But I should probably let Sen. Obama and his campaign address those kinds of questions.”

No candidate — and, make no mistake, Bayh has been actively campaigning for the vice-presidential nod — has done more to feed the speculation about himself than Bayh. And it’s working. Much of the national media seems to be embracing the notion that a joint Obama-Bayh appearance in Elkhart, Indiana, on Wednesday will unveil a ticket.

That would be the culmination of a long run by Bayh, who has been running for president or vice president for years. He’s been on Democratic VP short lists since 2000. And he made moves toward presidential 2004 and 2008 presidential bids before finally abandoning the notion.

Bayh got no traction as a presidential candidate because he hails from the Democratic Leadership Council wing of a party that has little taste — at least at the grassroots level where primaries and caucuses are decided — for the DLC’s “Republican-lite” politics.

While his not-so-Democratic ideology prevented him from making a serious bid for the presidency, the pro-Bayh line of argument now goes, this makes him an attractive running-mate for Obama in blue-collar states where the Illinois senator ran poorly in the primaries.

The problems with this line of “reasoning” are many.

For one thing, while Bayh is personally popular in Indiana, he has never been able to help Democrats win presidential races in a state where he has been the top Democrat for the better part of a quarter century. (The last Democrat to win a presidential election in Indiana was Lyndon Johnson, who was running in the days when Bayh’s father was serving in the Senate.)

Evan Bayh’s not even that good at uniting Indiana Democrats; while he campaigned ardently for Clinton in this year’s Hoosier primary, she barely won — a result that did serious damage to the New Yorker’s candidacy in the last weeks of the race.

Few serious observers suggest that adding Bayh to the ticket would guarantee Indiana for the Democrats this fall. But to avoid embarrassment, Bayh and Obama would be forced to concentrate on the state — drawing attention and energy away from more serious battlegrounds.

And what of the theory that, even if Bayh couldn’t deliver Indiana, he could help Obama win neighboring Ohio?

Well, let’s just say that argument is more popular with east-coast pundits who do not spend much time in the middle of the country than with actual Ohioans.

The notion that an Ohioan who was not going to vote for Obama would suddenly say, “Oh, he’s got a guy from Indiana on the ticket — that changes everything,” is comic.

The part of Ohio that borders Indiana is rigidly Republican. (The same goes for the parts of Michigan that border northern Indiana.) The parts of Ohio that decide presidential elections have no record of looking to Indiana in general, or Bayh in particular, for leadership. If Obama really wanted to carry Ohio — and Michigan — he would pick a populist with a better record than his own on trade issues. Instead of Bayh, he’d be looking at Ohio Senator Sherrod Brown.

Brown, who knows how to deliver an economic message that beats Republicans, could actually help Obama.

That’s more than can be said for Bayh who, like Joe Lieberman in 2000, threatens to balance the Democratic ticket in the wrong direction. This is why it is hard to believe that the Obama team, which usually does not make big mistakes, really sees the senator from Indiana as vice-presidential timber.

Of course, that’s not what Evan Bayh would have us believe.