American diplomatic and intelligence circles feared the Soviet Union even before the end of the Second World War. The whole world had witnessed Soviet power in defeating the German invasion of 1941-1945, and Americans feared that Soviet ideology coupled with Stalin's ambitions could prove an existential threat for the United States. Of course, Franklin Roosevelt did not distrust the Soviets so badly--he in fact envisioned a postwar world led by a continued alliance between the Soviet Union and the United States--but others, including Roosevelt's heir Harry Truman, were not so sure. So while, at the highest levels (Stalin and Roosevelt), Soviet-American relations only strengthened as the war approached its end, you can see fear manifest at lower levels; notably, a cooperative Soviet-American initiative to create common wartime propaganda fell to pieces after the Allies gained the clear advantage in the war (see Todd Bennett's One World, Big Screen: Hollywood, the Allies, and World War II, or at least the chapters on the Soviet Union).

With the passing of Roosevelt in April of 1945, relations began to deteriorate at the highest levels as well. Stalin feared that Truman would renege on agreements he made with Roosevelt, and so Stalin rushed to conquer the areas of Central Europe that Roosevelt had agreed at Yalta would reside in a "Soviet sphere of influence after the war." Truman--and we are getting to the meat of your question now--shortly thereafter hid from Stalin the fact that the United States had successfully tested an atom bomb. Truman chose to use it, too, without telling Stalin (and this further undermined Roosevelt's relations with Stalin, since Japan had been a focal point in their negotiations).

Fear came to dominate the American worldview after the war. In his famous "Long Telegram," diplomat George Keenan warned from Moscow in February of 1946 that the Russians could not be trusted. Their ideological desire to dominate all the world could only be contained, Keenen warned. That same year (I really am getting to your question now), the Central Intelligence Agency, one month after its founding, concluded in in October of 1946 that

It is probable that the capability of the USSR to develop weapons based on atomic energy will be limited to the possible development of an atomic bomb to the stage of production at some time between 1950 and 1953.

So the United States was always afraid of the Soviet Union. But Americans did not fear that the Soviet Union could keep up with, and potentially surpass the United States technologically until 1949, when the Soviet Union tested its first atomic bomb. The earliest the Americans thought they would have to contend with Soviet nuclear might was in the early 1950s, perhaps even a decade after the Americans had first tested their own nuclear device. Instead, the Americans faced another nuclear power just four years after they had created their own weapon. The Soviets were more advanced than the Americans thought, and American intelligence had failed to determine that fact.

Still, no moment compared to Sputnik. Up to that point, Americans still had the security of the oceans coupled with their industrial superiority. Even after the Soviet Union began producing its own nuclear bombs, Americans knew that they could win a nuclear war with the Soviets, even if at great cost. The United States could send in waves of bombers over the Soviet Union, while absorbing the bulk of the Soviet counterattack in Europe. Sputnik changes that formula, not only by ushering in the age of intercontinental ballistic missiles, but also by placing the Soviet Union at the head of that race.

I want to break here to underscore this point: a lot of people seem to miss that important detail regarding Sputnik. The satellite itself was almost useless; Sputnik scared Americans because of the rocket that was used to put it into space. It displayed a Soviet technology that the American didn't have! So now, if we were to head to a nuclear war, it was possible that the Soviets could build up an arsenal of ICBM's and destroy the United States.

Of course, the "Sputnik moment" ignited a passion for rocketry in the United States. By the time Kennedy became president in 1960, just a few years after Sputnik, he discovered that his fear mongering over a "missile gap" with the Soviet Union had been just that; the Americans had already taken back the lead and developed an arsenal of nuclear missiles that put the Soviets to shame. NASA, which Dwight Eisenhower had created in 1958 in response to Sputnik, took a rocket to the moon by 1969.

The Soviet Union never again took an American-identified lead in the nuclear arms race again, but the construction of a Soviet missile defense shield (as we call them today; back then, they were called "anti-ballistic missile systems") around Moscow underscored the fact that the Soviets would continue to prove a competent adversary in the race. This continued fear, mutual as it was, helped to bring about negotiations from 1968-1973 that ended the construction of missile defense shields, at least until Reagan reneged on that deal in the 1980s.

But, in short, Sputnik is the moment of peak fear of Soviet technology; 1957.

“We Now Know” by John Lewis Gaddis is an excellent book on the topic. Tangentially, “Skunk Works” by Ben Rich also fills in a lot of gaps, since the former book ends at the Cuban Missile Crisis.

The shortest answer is that the Eisenhower Administration knew. Eisenhower had made an hard push for aerial reconnaissance (and argued for an international “free skies” law on that topic, knowing full well that US lost nothing due to its geographic isolation). It became rapidly apparent that the USSR was deploying a lot of stagecraft to appear stronger than it was, and a great example of that was the so-called “bomber gap” that was a brief topic of intense political debate. At a 1955 air shoe, the Soviets had numerous M-4 Bison jet-powered bomber thundering overhead, leading analysts to think there were hundreds of such bombers ready to go, and the US was shockingly far behind. In truth, the Soviets had less that thirty, and we’re flying them in loops over the air show grounds. As the reconnaissance photos mapped out the USSR and discovered only a handful in the whole country, the fear of the “bomber gap” evaporated.

At the same time, Eisenhower was also aware he could afford to expose the Potemkin Village too nakedly, because even in their condition, destabilizing the Soviets was dangerous, and war would be ruinous (even though they could have only delivered a nuclear weapon by truck). Secondly, as a student of Clausewitz, he was familiar with the construct of “absolute war” and could see that nuclear weapons were close to that. So, his strategy was to try to keep everyone calm. Which was a fine enough thing - except that a lot of people in the US had a vested interest in stoking red dear into the US (cf. “the military industrial complex” speech).

And this is the game that would play out over the next decades. Apart from the occasional provocation (especially the Cuban Missile Crisis, where Khrushchev badly misjudged what American reaction would be) the Cold War was a game of Soviets trying to be a strong as they appeared and presidents trying not to humiliate them too badly while appearing concerned enough to fend off challenges at home.

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