How competitive would the Roman Empire at its height have been with medieval (11th-13th century) European armies in a war, both stuck at their understanding of warfare and technology?
The Roman army of Trajan would have crushed any medieval army, for one basic reason: the Roman army, unlike the levies and knights called together by medieval kings, was a professional fighting force. Its legions had training and - more importantly - financial and logistical resources that no medieval state could match. The Romans might have lost battles, but they would never have lost a war.More detail:First, the Roman Empire was much larger, and routinely fielded much larger armies, than any medieval state.Under Trajan, when the Empire probably had about 60 million inhabitants, there were about 300,000 in the Roman army (30 legions of c. 5,000 men each, each brigaded with a roughly equal number of auxiliaries).These were professional soldiers, enrolled for 25 years (and sometimes longer) and rigorously trained. They were issued high-quality weapons from state “factories,” were well-fed, and generally exhibited superb morale and esprit de corps.Let’s compare the French army (Medieval Europe’s largest) at the beginning of the Hundred Years’ War. [I realize the question said 11th-13th century, but I use mid-14th century France because we have more information]France in 1360 (the light red areas were ceded to England after the Battle of Poitiers)The French king Philip VI ruled an estimated 15–17 million subjects - more than any of his rivals. Yet at the Battle of Crecy (1346), the culminating moment of a major campaign to defend France from the invading army of Edward III of England, Philip was only able to muster 20–30,000 troops.Perhaps a third of this army consisted of mounted men-at-arms - knights with extensive military training (but often little trust for one another and less love for the king). With the exception of a large body of Genoese crossbowmen (more on the crossbow later), the remainder were ill-trained and poorly-organized peasant levies.Unless superbly commanded, this was not an army that could routinely win battles against a Roman force of equivalent size, let alone the Roman army at full strength.Medieval armies, however, had a few technological advantages….Although the basic means of war did not change much between the 2nd and the mid-14th centuries (guns existed by the 14th century, but were still too primitive to have much battlefield impact), the armies at Crecy used two technologies unknown to the Romans: the longbow (on the English side) and the stirrup.The Romans used both crossbows and composite bows (it is unclear in what proportion), typically, the heavy infantry of the legions was complemented by bands of auxiliary archers (and sometimes slingers).No Roman unit, however, was equipped with any weapon equivalent to the English longbow, which had a range (up to 350 yards) considerably greater than the (estimated) range of Roman bows.An even more important medieval development was the stirrup, which seems to have reached Europe around the 7th century CE.The Romans (as a previous poster noted) had heavy cavalry, particularly in late antiquity. In Trajan’s time, auxiliaries (Sarmatians and others) would have provided most of this arm. These horsemen, however, did not have stirrups, and were thus (despite high-cantled saddles) at risk of being thrown from their mounts when charging a line of infantry.Medieval knights, equipped with stirrups, delivered a more compelling shock to infantry armies than the Roman cataphracts on the initial charge, and had a much easier time staying in their saddles in the ensuing melee.The longbow and stirrup were real advances - but I doubt they would have been enough to give any but the largest and best-commanded medieval armies a chance of defeating a Roman force.The Romans, after all, knew how to deal with both heavy cavalry and archers. The second-century author Arrian explains how to defeat incursions of the nomadic Alans (a people of the Caspian steppes famed for their horsemen):“Once [the Roman troops are] thus arrayed there should be silence until the enemies come within missile range, when in range the loudest and most intimidating war cry must be raised by the whole lot, and bolts and stones must be fired from the artillery pieces and arrows from the bows, and javelins by both light armed and shield bearing javelinmen. Stones must also be thrown at the enemies by the allied force on the overwatch position, and the whole missile rain must be coming from all sides to make it concentrated enough to panic the horses and destroy the enemies. And the expectation is that the Scythians will not get close to the infantry battle formation because of the tremendous weight of missiles. If they do close in though, the first three ranks should lock their shields and press their shoulders and receive the charge as strongly as possible in the most closely ordered formation bound together in the strongest manner. The fourth rank will throw their javelins overhead and the first rank will stab at them and their horses with their spears without pause. After repulsing the enemy if there’s a clear rout, the infantry units must clear lanes and the horsemen should advance, not all squadrons, but only half of them. Those to the fore must be the first to advance. The other half should follow those that advance, in perfect formation and not in hot pursuit in order that they may continue the initial pursuit with fresh horses in case there is a complete rout, and in case they turn about to attack, they may assist those in pursuit. At the same time the Armenian archers must advance shooting their bows in order to prevent those in flight from turning about, and the light armed javelineers should advance at the run….”Finally, as noted at the beginning of this post, even if a medieval army managed to defeat the Romans in battle, it would almost certainly lose the war.In the reign of Trajan, an estimated 2/3 of the Roman Empire’s revenues were channelled directly to the army, even in times of peace. Soldiers were paid reliably, and paid well. They were also well-supplied. Local populations filled regular contracts to supply the legionary forts with meat, grain, and other provisions, and campaigns were carefully planned in advance to ensure that the troops were never hungry or thirsty.No medieval king had such resources. Feudalism created loosely-organized states, in which kings had to constantly struggle with their own nobles to secure the men and money needed for campaigning. If a parliament existed, the process was even more difficult. Medieval kings were thus forced to scrape armies together just before a major campaign, and to make their campaigns as short as possible, since they almost never had enough money to keep their troops in the field for long.The second-century Roman army, in short, would have won in almost any conceivable engagement with a medieval European force of remotely equivalent size - and since the Roman Empire was so much larger and better-organized than any medieval kingdom, it would, in my opinion, have easily overwhelmed any opponent before the widespread use of gunpowder.Those intrigued by the Roman army might want to check out my page on the Column of Marcus Aurelius.I wrote a follow-up answer to this one that might interest some readers:Garrett Ryan's answer to If the Roman armies were indeed superior to their medieval counterparts, why didn't the Eastern Roman Empire ever restore the empire fully (even beyond Justinian), for significant amount of time and later fall?