The seeds sprout after about a week of watering, with seedlings emerging from the top of the compost.
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These two carrot varieties are slightly different. The 'Rainbow' (in other words, mixture of heirloom coloured seeds) carrots grow about 6 inches long - I have harvested them at smaller lengths in shallower containers before and they still grew acceptably.
The second variety, which is known to be long and thin, grows about 8 inches long. The roots of this type are happy at this depth of the container. So, I do not think having more depth would interfere with the growth of these particular carrots. The most important part of this is having sufficient depth so they can grow to their intended size.