In the old British standard we use a time dependent modulus of elasticity of concrete to calculate crack width, which is usually about half of the 28-day Ec for long term modulus. In Eurocode EN 1992-1-1:2004, from clause 3.1.3, it seems that Ecm is defined as the 28-day modulus of elasticity of concrete.
Is this definition correct, or should the formula be interpreted as Ecm(t) instead?
There are two major effects that make the elasticity modulus of concrete vary with time: hardening and creep.
In a Eurocode context $E_{cm}(t)$ refers to the hardening of concrete until it reaches its 28-day strength. This is section 3.1.3(3). If you need to calculate a crack width at an early stage, you should probably use $E_{cm}(t)$ instead of $E_{cm}$, but this has nothing to do with the long term modulus.
Creep or long term effects is not included in $E_{cm}$ (refer for example section 3.1.4(3)). For equation this is accounted for by the factor $k_t$ and by the fact that you should account for creep in the calculation of the stresses in the cross section. This might be the confusing part: Even though equation (7.9) uses an elasticity modulus equal to $E_{cm}$, the steel stress in the same equation should be calculated using an elasticity modulus which is not equal to $E_{cm}$ but corrected for creep in accordance with section 3.1.4(3). Unless there are no long term loads, of course, in which case it gets a little simpler.
