log$_{10}$ κ M/ 10 $^{16}$GeV F (A) − 2 . 79 ± 0 . 30 0 . 450 ± 0 . 062 F (B) − 1 . 94 ± 0 . 34 0 . 563 ± 0 . 038 D (A) − 4 . 55 ± 0 . 21 0 . 196 ± 0 . 030 D (B) − 3 . 84 ± 0 . 23 0 . 330 ± 0 . 047 CMB +SDSS +BBN +SDSS/BBN τ = 0 -21.7 -20.4 -20.8 -20.2 HZ -9.4 -11.8 -14.6 -16.8 F (A) -1.2 -4.4 -10.2 -13.2 F (B) 1.2 0.2 -4.0 -5.6 D (A) -4.0 -8.6 -14.2 -17.6 D (B) 0.2 -2.8 -9.6 -12.2 TABLE II: Δ χ = − 2 log( L$_{1}$ / L$_{2}$ ) for minimal 6 parameter F and D -term scenarios. These are defined as model 2; model 1 is the standard 6 parameter power law. We also present results relative to the 5 parameter τ = 0 and HZ models. TABLE I: Constraints on minimal F and D -term parameters using string models A and B. the data. This is because the best-fitting D-term models and those with n$_{S}$ = 1, in the absence of the BBN prior, typically favour values of Ω$_{b}$ h 2 ∼ 0 . 024 − 0 . 025. The situation is more complicated in the F-term case. In both cases the F-term model is disfavoured relative to the 6 parameter model, but for string spectrum B the Δ χ 2 is only − 5 . 6. This would not be enough to exclude the model with high confidence, whereas for spectrum A the Δ χ 2 = − 12 . 2, which would. We have already alluded to the reason for this: spectrum B allows for larger values of n$_{S}$ than the spectrum A. In order to calibrate our interpretation of these values we have done two things. First, we computed Δ χ 2 for a 5 parameter model with τ = 0. For each combination of data, we find Δ χ 2 ≈ − 20, with roughly 5 σ signifi- cance that τ = 0 (for example, CMB data alone gives τ = 0 . 090 ± 0 . 017). We have also done the same for models with n$_{S}$ = 1, that is a Harrison-Zel’dovich (HZ) spectrum. For CMB+SDSS data we find Δ χ 2 = − 11 . 8, corresponding to the HZ model being excluded at 3 . 4 σ , and for CMB+SDSS+BBN data Δ χ 2 = − 16 . 8, corre- sponding to 4 . 1 σ ( n$_{S}$ = 0 . 951 ± 0 . 012). These values are presented in table II. Comparing the Δ χ 2 for the F- and D-term models we can conclude that the D-term model is excluded at a level somewhere in the range of 3-4 σ even if strings are included, with the precise level dependent on which string spectrum is correct. Meanwhile the F-term model would be excluded at around 3 σ if string spectrum A is correct, but cannot be excluded is spectrum B is correct. The statistical strength of our results is very strongly related to the value of D/H measured in ref. [12] and the subsequent interpretation in terms of BBN. In order to explore the effects of systematic offsets in the value of Ω$_{b}$ h 2 on our results we have computed the value of Δ χ 2 for a prior of ω$_{b}$ = Ω$_{b}$ h 2 = ¯ ω$_{b}$ ± 0 . 001 (i.e. a different central value with the same statistical error as ref. [12]). For the D-term model with string spectrum A, the vari- ation in χ 2 for CMB+SDSS+BBN can be approximated by Δ χ 2 ≈ 3160 ¯ ω$_{b}$ − 84 . 5. Even if the central value were to increase by 0.002, the model would still be in tension with the data, Δ χ 2 ≈ − 11. Non-minimal models: Non-minimal F- and D-term models can be considered; these typically involve the in- clusion of a negative mass-square term in the potential and can allow for values of n$_{S}$ lower than 0.98 due to an enhanced negative curvature of the potential. In addi- tion, at the point of horizon exit, the potential can be effectively flattened. This leads to a lower energy-scale of inflation and therefore to lower values for M and a smaller string tension. In ref. [7] we considered a par- ticular class of F-term models [13] characterized by the addition of the potential term c 2 $_{H}$H $^{2}$S$^{2}$, where H denotes the Hubble rate. This “Hubble-dependent” mass term generically arises in the presence of a non-minimal K¨ ahler potential from the F-term that drives inflation. In the absence of F-terms, even for a non-minimal K¨ ahler potential, no such additional mass-square correc- tion can occur. This is why for the simplest realizations of D-term inflation, the exclusion of mass-square correc- tions is well-motivated. However, there are particular models where sub-dominant F-terms ( e.g. from compact- ification in string models or from sneutrino potentials), that do not drive inflation, may be present [14–16]. These can lead to mass-square corrections for the D-term po- tentials that may redden the spectrum and reduce the string contribution to the CMB. We aim to formulate the model in such a way that the number of parameters remains small ( e.g no more than 7 parameters) and presume strings form. This can be achieved by the restriction to g < ∼ 0 . 1 and adding the potential term in the form of g $^{2}$c2 $_{H}$S $^{2}$. This way, the predictions remain independent of the gauge coupling g . We note that in presence of a negative mass-square term, there may also be models with g > ∼ 0 . 1, that are no longer disfavoured by the data. Unless one applies a restriction to more specific models, a systematic study of this region of parameter space would require additional free param- eters in the analysis and is therefore not performed here. In Fig. 2 we show 2D likelihoods for the non-minimal models using string spectrum A. In the F-term case, neg- ative values of c 2 H lead to an increased red-tilt in n$_{S}$ , al- leviating the tension with data present for c 2 H = 0. The fit is slightly better than the 6 parameter model with Δ χ 2 = 1 . 2. We find marginalized values of log$_{10}$ κ = − 2 . 28 ± 0 . 46, M = (0 . 346 ± 0 . 065) × 10 $^{16}$GeV and c 2 H = − 0 . 027 ± 0 . 011. For the D-term model the situation is more compli- cated, since for fixed P$_{R}$ and κ the potential can admit two solutions, each with different M and parameters n$_{S}$ and Gµ . In the analysis we treat these independently, running separate MCMC chains for each solution branch. The results of the analysis are shown in Fig. 2. The 3