Decisions are made by those who turn up. This law applies ruthlessly in politics as the Labour party and the young were brutally reminded in May 2015. Had the result of our most recent general election been determined by the under-25s, Ed Miliband would now be prime minister. Had the outcome been decided by the under-45s, it would have been a dead heat between the two major parties. As it was, Labour limped in seven points and nearly two million votes behind the Conservatives because older cohorts of the electorate leant heavily to the Tories and grandpa and grandma turned up at the polling stations in the largest numbers. When David Cameron walked back into Number 10, the keys were handed to him by silver voters.

If the same happens on 23 June, if youngsters stay at home while oldsters swarm to the polling stations, then it is much more likely that Britain will be taking its leave of the European Union. Who turns up for the referendum will be decisive to the outcome. If the battle is close, it will make all the difference. Close it currently appears to be. The Opinium poll that we publish today gives a narrow lead to the Outers. I issue my usual health warning that we should not read too much into the headline numbers of one poll. It is also important to note that Opinium conducts its surveys online. Internet polls are tending to come up with better scores for the Outs while phone polls on the referendum are generally more encouraging for the Ins.

What really leaps out of the data is the striking contrast about attitudes towards Europe between the generations. Among the under-35s, it is a landslide for staying in. Remain prevails over leave by 53% to 29%. The figures are almost exactly reversed among those aged 55 and over. They are 54% to 30% for quitting the EU. It is neck and neck among the middle-aged. That’s in line with surveys by other polling companies, which also suggest that attitudes to Europe are highly correlated with age. The fewer wrinkles you have, the much more likely it is that you want to remain in the EU; the thinner your hair, the much more likely you are to yearn for Brexit. A very generalised – but probably accurate – way of explaining the difference is that this younger generation is better educated and more travelled than any before it. You could also see it as an inversion of usual stereotypes about age. It is wild oldsters who want to take a leap into the unknown by self-ejecting from the world’s largest trading area and it is level-headed youngsters who prefer the security of the status quo.

What makes the generation gap potentially very dangerous for the In campaign is that oldsters say they are much more likely to turn up on the day of decision. More than 80% of the over-55s declare that they are absolutely certain to vote in the referendum. Barely more than half of the under-35s say the same. The actual differential may be wider and worse than that. One of the reasons that the opinion pollsters got the general election so wrong is that their samples included too many young voters who said they were definitely going to cast a ballot and then proved to be no-shows on the day.

For all the breath that the prime minister has expended telling people that this may be the most important vote they ever cast, the message is clearly not getting through to many of the young. It probably isn’t helping that the most prominent faces of the rival campaigns are middle-aged and male, Westminster, corporate and other establishment figures on the In side and even older and even more male suits on the Out side.

Younger voters are more likely to identify with Labour than the Tories. So it is surely an additional problem that the Labour leadership is radiating so little enthusiasm for mounting a vigorous campaign for an In vote with the result that the debate on the airwaves is being dominated by arguments between rival flavours of Conservative. Our poll finds that large majorities of people can correctly identify which sides of the battle David Cameron and Boris Johnson are on, but 40% of voters say they still don’t know whether Jeremy Corbyn is for In or for Out. Although Alan Johnson is the chairman of the Labour In campaign, just 25% of people say they know which team he belongs to.

“We are acutely aware of the generation gap,” says one strategist on the In team. Aware and clearly quite anxious about it. They are trying to bridge the gap with a lot of effort, especially concentrated on university campuses, where they now say they have more than 50 In groups established. There are also strong indications that David Cameron is becoming anxious that the oldsters who returned him to power last year will bring a humiliating end to his premiership in three months’ time and defeat him in the referendum if the young don’t turn up.

One sign of those nerves was the recent speech by Nicky Morgan, the pro-remain education secretary, in which she implored young people to vote on the grounds that they would “suffer the most” if Britain amputates itself from the EU.

Both campaigns believe that peer groups and families are huge influencers on how people will vote. So Mr Cameron recently exhorted older people to think about their children and grandchildren before they voted to leave the EU and Ms Morgan urged young people “to go out and make the case to others and in particular your older friends and relatives”. Trying to use the young to change the minds of older relatives is an idea stolen from last year’s referendum in Ireland on same-sex marriage. There was a huge difference in generational attitudes at the outset of that campaign with younger voters heavily in favour of legalising gay marriage and older voters heavily against. The ultimate and large victory for the yes side in Ireland is often credited to young people persuading their parents and grandparents to change their minds.

For that sort of tactic to have any chance of success in the EU referendum, first the young themselves have to be engaged with the arguments, registered to vote and enthusiastic about going to the polls. On all three scores, there are reasons for the In team to be anxious.

One worry for them is the impact of the switch from household to individual voter registration. An estimated 800,000 people have evaporated from the electoral roll. They are disproportionately urban-dwelling and mobile with students in university cities and towns at particularly high risk of losing their vote. Those missing voters are most likely to be young and pro-membership of the EU.

A related challenge for the In side is that, while the young may be much keener to remain within the EU than their grandmothers and grandfathers, they are also much less animated about Europe as an issue. More than half of the over-65s regard EU membership as one of the top three issues facing Britain. Barely more than a tenth of 25 to 34-year-olds put it in their top three.

What you might call the differential passion between the two sides is probably the greatest handicap for the Ins and the Outers’ clearest path to victory. The older voter who hates the EU is pretty certain to express their feelings at the ballot box in June. On the other side, there are few people who love the EU in a way that matches the intensity with which the phobes loathe it. This is related to another difference between the generations. Research for the Electoral Reform Society finds that older voters are much more likely than younger ones to say that they feel well-informed about the choice.

There is also a gender gap. Men are twice as likely to say they feel well-informed as women. This does not mean that older men actually have a better grasp of the facts; just that older men are more likely to think that they know it all. There seems to be a link between how much people feel they understand about the choice and how likely they are to vote. Which gives the In campaign something else to worry about. The ERS recommends that both campaigns should commit to making voter registration a key plank of their campaigns, which is a good idea. It also suggests that there should be a “ceasefire week” during which both sides commit to laying off the negative stuff and presenting only the positive sides of each case. That is a sweet idea. Too sweet to be likely to happen.

When Britain makes this enormous choice on 23 June, the outcome will matter most to the young because they will have to live with its consequences for far longer than their parents and grandparents. That should be obvious, but at the moment it doesn’t seem to be anything like sufficiently appreciated. If the young don’t turn up on the day of decision, if they are once again swamped at the ballot box by silvery voters, they will hand their destinies to the old. The young should vote as if their future depended on it, because it does. You can register to vote online. It costs nothing except five minutes of your time. And you can do it here: gov.uk/register-to-vote