However, our uncertainty about the present average density of the universe is even greater. If we add up the masses of all the stars that we can see in our galaxy and other galaxies, the total is less than one hundredth of the amount required to halt the expansion of the universe, even for the lowest estimate of the rate of expansion. Our galaxy and other galaxies, however, must contain a large amount of “dark matter” that we cannot see directly, but which we know must be there because of the influence of its gravitational attraction on the orbits of stars in the galaxies. Moreover, most galaxies are found in clusters, and we can similarly infer the presence of yet more dark matter in between the galaxies in these clusters by its effect on the motion of the galaxies. When we add up all this dark matter, we still get only about one tenth of the amount required to halt the expansion. However, we cannot exclude the possibility that there might be some other form of matter, distributed almost uniformly throughout the universe, that we have not yet detected and that might still raise the average density of the universe up to the critical value needed to halt the expansion.
The present evidence therefore suggests that the universe will probably expand forever, but all we can really be sure of is that even if the universe is going to recollapse, it won’t do so for at least another ten thousand million years, since it has already been expanding for at least that long. This should not unduly worry us: by that time, unless we have colonized beyond the Solar System, mankind will long since have died out, extinguished along with our sun!
All of the Friedmann solutions have the feature that at sometime in the past (between ten and twenty thousand million years ago) the distance between neighboring galaxies must have been zero. At that time, which we call the big bang, the density of the universe and the curvature of space-time would have been infinite. Because mathematics cannot really handle infinite numbers, this means that the general theory of relativity (on which Friedmann’s solutions are based) predicts that there is a point in the universe where the theory itself breaks down. Such a point is an example of what mathematicians call a singularity.
Halt 停止
Uniformly 一致地
Unduly 過度地
Singularity 奇點
然而，我們對現在宇宙的平均密度測量得更不準。我們如果將銀河系和其他所有能看到的星系的恆星的質量加起來，甚至是按對膨脹率的最低的估值而言，其質量總量比用以阻止膨脹的臨界值的1％還少。然而，在我們以及其他的星系裡應該有大量的“暗物質”，那是我們不能直接看到的，但由於它的引力對星系中恆星軌道的影響，我們知道它必定存在。況且人們發現，大多數星系是成團的。類似地，由其對星系運動的效應，我們能推斷出還有更多的暗物質存在於這些成團的星系之間。將所有這些暗物質加在一起，我們仍只能獲得必須用以停止膨脹的密度的1/10。然而，我們不能排除這樣的可能性，可能還有我們未能探測到的其他的物質形式幾乎均勻地分佈於整個宇宙，它仍可以使得宇宙的平均密度達到停止膨脹所必要的臨界值。所以，現在的證據暗示，宇宙可能會無限地膨脹。但是，所有我們能真正瞭解的是，既然它已經膨脹了100億年，即便如果宇宙還要坍縮，則至少要再過這麼久才有可能。這不應使我們過度憂慮——到那時候。除非我們到太陽系以外開拓殖民地，人們早由於太陽的熄滅而死亡殆盡！
所有的弗利德曼解都具有一個特點，即在過去的某一時刻（約100到200億年之前）鄰近星系之間的距離為零。在這被我們稱之為大爆炸的那一時刻，宇宙的密度和空間——時間曲率都是無窮大。因為數學不能處理無窮大的數，這表明廣義相對論（弗利德曼解以此為基礎）預言，在宇宙中存在一點，在該處理論自身失效。這正是數學中稱為點的一個例子。
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