Monte Carlo Markov chains analysis of WMAP3 and SDSS data points to broken symmetry inflaton potentials and provides a lower bound on the tensor to scalar ratio

We perform a Monte Carlo Markov chains (MCMC) analysis of the available cosmic microwave background (CMB) and large scale structure (LSS) data (including the three years WMAP data) with single field slow-roll new inflation and chaotic inflation models. We do this within our approach to inflation as an effective field theory in the Ginsburg-Landau spirit with fourth degree trinomial potentials in the inflaton field $\varphi$. We derive explicit formulae and study in detail the spectral index $n_s$ of the adiabatic fluctuations, the ratio $r$ of tensor to scalar fluctuations, and the running index $d{n}_s/d\ln k$. We use these analytic formulas as hard constraints on $n_s$ and $r$ in the MCMC analysis. Our analysis differs in this crucial aspect from previous MCMC studies in the literature involving the WMAP3 data. Our results are as follows: (i) The data strongly indicate the breaking (whether spontaneous or explicit) of the $\varphi \to -\varphi$ symmetry of the inflaton potentials both for new and for chaotic inflation. (ii) Trinomial new inflation naturally satisfies this requirement and provides an excellent fit to the data. (iii) Trinomial chaotic inflation produces the best fit in a very narrow corner of the parameter space. (iv) The chaotic symmetric trinomial potential is almost certainly ruled out (at 95% C.L.). In trinomial chaotic inflation the MCMC runs go towards a potential in the boundary of the parameter space and which resembles a spontaneously symmetry broken potential of new inflation. (v) The above results and further physical analysis here lead us to conclude that new inflation gives the best description of the data. (vi) We find a lower bound for $r$ within trinomial new inflation potentials: $r>0.016(95\%\mathrm{CL})$ and $r>0.049(68\%\mathrm{CL})$. (vii) The preferred new inflation trinomial potential is a double well, even function of the field with a moderate quartic coupling yielding as most probable values: $n_s\simeq 0.958$, $r\simeq 0.055$. This value for $r$ is within reach of forthcoming CMB observations.

Figure 1

Trinomial Chaotic Inflation.—We plot here the chaotic inflation trinomial potential [Eq. (3.2) with positive quadratic term] $\frac{y}{8}w(\chi )$ vs the field variable $\sqrt{z}=\sqrt{y/8}\chi$ for different values of the asymmetry parameter $h$, namely, $h=0$, $-0.5$, $-0.8$, and $-1$. Notice the inflection point at $\sqrt{z}=1$ when $h=-1$.

Figure 2

Trinomial Inflation.—We plot $r$ vs $n_s$ for fixed values of the asymmetry parameter $h$ and the field $z$ varying along the curves. The two black curves on the left ($h=0$ and $h=20$) are for new inflation. All other red curves on the right are those of chaotic inflation with $-0.99\le h\le 0.99$ (only the short magenta curve in the top has positive $h$). The filled areas correspond to 12%, 27%, 45%, 68%, and 95% confidence levels according to the WMAP3 and Sloan data. The filling of the areas goes from the darker to the lighter for increasing C.L. New inflation only covers a narrow area between the black lines on the left while chaotic inflation covers a much wider area but, as shown by Fig. 9, this wide area is only a small corner of the field $z$-asymmetry $h$ plane. Since new inflation covers the banana-shaped region between the black curves, we see from this figure that the most probable values of $r$ are definitely nonzero within trinomial new inflation. Precise lower bounds for $r$ are derived from MCMC in Eq. (5.2).

Figure 3

Trinomial New Inflation.—The new inflation trinomial potential $\frac{yw(\chi )}{8(h^2+1{)}^2}$ [Eq. (4.2)] vs the field variable $\frac{\sqrt{y}\chi }{\sqrt{8}\sqrt{h^2+1}}$ for different values of the asymmetry parameter $h=0$, $-0.4$, $-1$, $-20$. We have normalized here the field variable and the potential by $h$-dependent factors in order to have a smooth $|h|\gg 1$ limit.

Figure 4

Comparison of the marginalized probability distributions (normalized to have maximum equal to one) of the most relevant cosmological parameters (both primary and derived) between trinomial new inflation (solid blue curves) and the $\Lambda \mathrm{CDM}+r$ model (dashed red curves).

Figure 5

Upper panels: 95% and 68% percent contour plots of joint probability $(\tau ,n_s)$ distribution (left) and $(\tau ,r)$ distribution (right) in the $\Lambda \mathrm{CDM}+r$ model. Lower panels: the two joint distributions in trinomial new inflation. Recall that in this case $n_s<0.9615\dots$ is a theoretical bound when $N=50$.

Figure 6

Trinomial New Inflation.—Upper panels: probability distributions (solid blue curves) and mean likelihoods (dot-dashed red curves), all normalized to have maximum equal to one, for the values of the normalized coupling at horizon exit $z_1$, of the quartic coupling $y$ and of the modulus $|h|$ of the asymmetry of the potential. Lower panels: probabilities and mean likelihoods for the values of $n_s$ and $r$. Notice that here $n_s<0.9615\dots$ and $r\le 0.16$ since these are the theoretical upper bounds for $n_s$ and $r$ in trinomial new inflation with $N=50$. Trinomial new inflation exhibits a lower bound for the ratio: $r>0.016(95\%\mathrm{CL})$, $r>0.049(68\%\mathrm{CL})$.

Figure 7

Comparison of the marginalized probability distributions (normalized to have maximum equal to one) of the most relevant cosmological parameters (both primary and derived) between trinomial chaotic inflation (solid blue curves) and the $\Lambda \mathrm{CDM}+r$ model (dashed red curves).

Figure 8

Trinomial Chaotic Inflation.—Upper panels: probability distributions (solid blue curves) and mean likelihoods (dot-dashed red curves), all normalized to have maximum equal to one, for the values of the normalized coupling at horizon exit $z$, of the quartic coupling $y$, and of the asymmetry $h$ of the potential. Lower panels: probabilities and mean likelihoods for the values of $n_s$ and $r$. The data request a strongly asymmetric potential in chaotic inflation. That is, a strong breakdown of the $\chi \to -\chi$ symmetry.

Figure 9

Trinomial Chaotic Inflation.—12%, 27%, 45%, 68%, and 95% confidence levels of the probability distribution in the field $z$-asymmetry $h$ plane. We see a strong preference for a very asymmetric potential with $h<-0.95$ and a significative nonlinearity $0.7<z<1$ in chaotic inflation.