In this article it is written ("the following" in the question above):
"In the Casimir effect, two flat plates placed very closely together restrict the wavelengths of quanta which can exist between them. This, in turn, restricts the types and hence number and density of virtual particle pairs which can form in the intervening vacuum and can result in a negative energy density. This causes an attractive force between the plates, which has been measured."
Now if the vacuum has (supposedly) a very high energy density, how can it be that at the same time, in certain situations, it has a negative energy density? This very high (constant in time) energy density, which is supposed to keep mass in the Universe together (though in a different way negative vacuum pressure pushes mass apart by increasing the space between them: the positive energy density doesn't keep mass together by decreasing the distance between them but by a very high curvature), increases the total amount of vacuum energy while the Universe expands. This doesn't contradict the conservation of energy because the negative pressure of the vacuum, does negative work in relation to the expanding Universe so the increase of the positive energy, is (miraculously?) canceled out by this negative work (energy). I find the situation around the vacuum energy somehow confusing. Some say it's zero, others it's enormous, is the vacuum energy the combination of the work done by the negative pressure and the positive vacuum density, is the vacuum energy maybe zero as is the pressure? Experiments that have shown that the vacuum energy density is close to zero (and positive), which makes it all even more confusing.
Can it be that above quote (about the vacuum having a negative energy density) has something to do with what I just wrote?
The bottom line: Why says the quote from the article a negative energy density can result, while calculations in the framework of the standard model show it's a very huge (something like a 1 whit 120 zeros behind it) energy density?
P.S. In connection with my question, this is a very interesting piece to read.