What is the large band of muscle that controls the size of the chest cavity called?

DIAPHRAGM. Actually the diaphragm is strongly attached to the base of sternum, then spine and also the ribcage at lower parts, So when diaphragm contracts, the length gets increases and also the diameter of chest, which expands the lungs, where the intercostal muscles controls and helps to move the ribcage and support in breathing, hence controls the size of the chest cavity.