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Population synthesis studies constitute a powerful method to reconstruct thebirth distribution of periods and magnetic fields of the pulsar population.When this method is applied to populations in different wavelengths, it canbreak the degeneracy in the inferred properties of initial distributions thatarises from single-band studies. In this context, we extend previous works toinclude $X$-ray thermal emitting pulsars within the same evolutionary model asradio-pulsars. We find that the cumulative distribution of the number of X-raypulsars can be well reproduced by several models that, simultaneously,reproduce the characteristics of the radio-pulsar distribution. However, evenconsidering the most favourable magneto-thermal evolution models with fastfield decay, log-normal distributions of the initial magnetic fieldover-predict the number of visible sources with periods longer than 12 s. Wethen show that the problem can be solved with different distributions ofmagnetic field, such as a truncated log-normal distribution, or a binormaldistribution with two distinct populations. We use the observational lack ofisolated NSs with spin periods P>12 s to establish an upper limit to thefraction of magnetars born with B > 10^{15} G (less than 1\%). As futuredetections keep increasing the magnetar and high-B pulsar statistics, ourapproach can be used to establish a severe constraint on the maximum magneticfield at birth of NSs.

Population Synthesis of Isolated Neutron Stars with magneto-rotational evolution II: from radio-pulsars to magnetars