Correlated primordial perturbations in light of CMB and large scale structure data

We use cosmic microwave background (CMB) and large scale structure data to constrain cosmological models where the primordial perturbations have both an adiabatic and a cold dark matter (CDM) isocurvature component. We allow for a possible correlation between the adiabatic and isocurvature modes, and for different spectral indices for the power in each mode and for their correlation. We do a likelihood analysis with 11 independent parameters and discuss the effect of choosing the pivot scale for the definition of amplitude parameters. The upper limit to the isocurvature fraction is 18% around a pivot scale $k=0.01{\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$. For smaller pivot wavenumbers the limit stays about the same. For larger pivot wavenumbers, very large values of the isocurvature spectral index are favored, which makes the analysis problematic, but larger isocurvature fractions seem to be allowed. For large isocurvature spectral indices $n_{\mathrm{i}\mathrm{s}\mathrm{o}}>2$ a positive correlation between the adiabatic and isocurvature mode is favored, and for $n_{\mathrm{i}\mathrm{s}\mathrm{o}}<2$ a negative correlation is favored. The upper limit to the nonadiabatic contribution to the CMB temperature variance is 7.5%. Of the standard cosmological parameters, determination of the CDM density ${\omega }_c$ and the sound horizon angle $\theta$ (or the Hubble constant $H_0$) are affected most by a possible presence of a correlated isocurvature contribution. The baryon density ${\omega }_b$ nearly retains its “adiabatic value”.

Figure 1

The unit-amplitude component angular power spectra ${\widehat{C}}_l^{\mathrm{a}\mathrm{d}}$ (dash-dotted line), ${\widehat{C}}_l^{\mathrm{i}\mathrm{s}\mathrm{o}}$ (dotted line), and ${\widehat{C}}_l^{\mathrm{c}\mathrm{o}\mathrm{r}}$ (solid line) of Eqs. (13, 20) for the case of spectral indices $n_{\mathrm{a}\mathrm{d}}=n_{\mathrm{i}\mathrm{s}\mathrm{o}}=1$ and other cosmological parameters representing median values of their marginalized likelihoods from our 11-parameter model. These curves would represent the relative contributions to the total $C_l$ for the case $\alpha =0.5$, $\gamma =1$, i.e., “equal” weights for the adiabatic and isocurvature contributions and a maximal positive correlation between them.

Figure 2

The same as Fig. (1), but for (a) ${\widehat{C}}_l^{\mathrm{T}\mathrm{E}}$ and (b) the matter power spectrum $\widehat{P}(k)$. We also show (c) the ${\widehat{C}}_l^{\mathrm{T}\mathrm{T}}$ of Fig. 1 with a logarithmic scale, so that the effect of changing the spectral indices can be readily estimated from the figure. The pivot scale $k_0=0.01{\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ becomes $k_0/h=0.01418{\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ for the parameter values used ($h=0.7053$) for this plot.

Figure 3

Marginalized likelihood functions for the standard cosmological parameters (i.e. those that exist in the adiabatic model). The (solid) line is the likelihood in our 11-parameter model, the (dashed) line is for the adiabatic model. Other line types show the effects of additional priors discussed in the text: (dotted) for Gaussian ${\Omega }_{\Lambda }=0.70\pm 0.04$, and (dot-dashed) for Gaussian $\gamma =0.0\pm 0.02$.

Figure 4

Marginalized likelihoods for two derived parameters, ${\Omega }_{\Lambda }$ and $H_0$. The line styles have the same meaning as in Fig. 3.

Figure 5

Marginalized likelihoods for the parameters related to the isocurvature mode and correlation. We show the $4$ remaining independent parameters, $\alpha$, $\gamma$, $n_{\mathrm{i}\mathrm{s}\mathrm{o}}$, $n_{\mathrm{a}\mathrm{d}2}$. The (solid) line is the full likelihood, other line types show the effects of additional priors discussed in the text: (dotted) for Gaussian ${\Omega }_{\Lambda }>0.82$, (dashed) for Gaussian ${\Omega }_{\Lambda }=0.70\pm 0.04$, and (dot-dashed) for Gaussian $\gamma =0.0\pm 0.02$.

Figure 6

Marginalized likelihoods for the isocurvature-related derived parameters, ${\alpha }_{\mathrm{c}\mathrm{o}\mathrm{r}}$ and $n_{\mathrm{c}\mathrm{o}\mathrm{r}}$. The line styles have the same meaning as in Fig. 5.

Figure 7

(a) Marginalized likelihoods for the effective adiabatic spectral index $n_{\mathrm{a}\mathrm{d}}^{\mathrm{e}\mathrm{f}\mathrm{f}}$ (solid) compared to the spectral index of the pure adiabatic model $n_{\mathrm{a}\mathrm{d}}$ (dashed). (b) Marginalized likelihood for $d{n}_{\mathrm{a}\mathrm{d}}^{\mathrm{e}\mathrm{f}\mathrm{f}}/d\ln \overline{k}$ at the pivot scale $k_0=0.01{\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$.

Figure 8

(a) 2-d marginalized likelihood for $\theta$ and ${\omega }_c$, showing that the large sound horizon angles $\theta$ are connected with low CDM densities ${\omega }_c$. We indicate the 68% (solid) and 95% (dashed) C.L. regions for our isocurvature model and for the adiabatic model. (b) 2-d likelihood for the two derived parameters $H_0$ and ${\Omega }_{\Lambda }$ closely related to the independent parameters $\theta$ and ${\omega }_c$.

Figure 9

2-d marginalized likelihood for ${\alpha }_{\mathrm{c}\mathrm{o}\mathrm{r}}$ and ${\Omega }_{\Lambda }$, showing that the large ${\Omega }_{\Lambda }$ are connected with a positively correlated isocurvature contribution.

Figure 10

(a) The TT spectrum for the best-fit large ${\Omega }_{\Lambda }({\Omega }_{\Lambda }>0.82)$ model, showing how the correlation part contributes towards shifting the acoustic peaks to the right. This model has ${\omega }_b=0.0221$, ${\omega }_c=0.100$, $\theta =1.082$, $\tau =0.177$, $b=0.796$, $\ln ({10}^{10}{A}^2)=3.24$, $n_{\mathrm{a}\mathrm{d}1}=0.949$, $\alpha =0.108$, $n_{\mathrm{i}\mathrm{s}\mathrm{o}}=3.11$, $\gamma =0.57$, $n_{\mathrm{a}\mathrm{d}2}=1.043$. The correlated adiabatic component (ad2, dash-dotted line) dominates over the uncorrelated adiabatic component (ad1, lower dash-dotted line in this figure). The other curves are the correlation component (cor, lower solid line) and isocurvature component (iso, dotted line). The total angular power (tot, solid line) is a sum of these components. (b) The matter power spectrum, showing how the correlation contribution changes its shape.

Figure 11

The 68% (solid) and 95% (dashed) C.L. regions in the $({\omega }_b,H_0)$ plane for our isocurvature model and for the adiabatic model. The rectangular gray boxes represent a BBN constraint ${\omega }_b=0.020\pm 0.002$ [37] and the HST constraint $H_0=(72\pm 8)$ km/s/Mpc [48].

Figure 12

2-d marginalized likelihood for $\alpha$ and $n_{\mathrm{i}\mathrm{s}\mathrm{o}}$.

Figure 13

Like Fig. 2c, but now with $n_{\mathrm{i}\mathrm{s}\mathrm{o}}=2.252$.

Figure 14

2-d marginalized likelihood for ${\alpha }_{\mathrm{c}\mathrm{o}\mathrm{r}}$ and $\alpha$.

Figure 15

2-d marginalized likelihood for ${\alpha }_{\mathrm{c}\mathrm{o}\mathrm{r}}$ and $n_{\mathrm{c}\mathrm{o}\mathrm{r}}$.

Figure 16

2-d marginalized likelihood for $n_{\mathrm{a}\mathrm{d}1}$ and $n_{\mathrm{a}\mathrm{d}2}$.

Figure 17

The CMB and matter power spectra for our best-fit model. The correlated adiabatic component (ad2) dominates over the uncorrelated adiabatic component (ad1) and the negative correlation (cor, dashed line shows negative values) subtracts from the Sachs-Wolfe region of the CMB spectrum.

Figure 18

The matter power spectrum of a model with $n_{\mathrm{i}\mathrm{s}\mathrm{o}}=5.69$, which has ${\chi }^2=1459.20$. The inset shows the SDSS window function for the 16th, i.e., second from right, data point. The markers ($*$) indicate the values (theoretical matter power spectrum convolved with the window function) to be compared to the data points, showing that the fit (${\chi }^2$) obtained this way is very good, although the power spectrum seems to lie below the data. The problem with this kind of matter power is that the window function picks most of the “power” from the nonlinear regime. Moreover, we have calculated the matter power up to $k/h\sim 3{\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ only. Had we calculated further, the rapidly rising power at large $k$ would have caused the markers ($*$) to move up leading to worse ${\chi }^2$ for this model.

Figure 19

Marginalized likelihoods of nonadiabatic contributions to the observed spectra in our 11-parameter model.

Figure 20

Marginalized likelihoods of nonadiabatic contributions to the observed spectra in “uncorrelated models” ($\gamma =0.0\pm 0.02$).

Figure 21

Marginalized likelihoods for ${\omega }_c$, $\theta$, $\alpha$, $n_{\mathrm{i}\mathrm{s}\mathrm{o}}$, $\gamma$, and $n_{\mathrm{a}\mathrm{d}2}$, using three different pivot scales, $k_0=0.002{\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ (dashed), $k_0=0.01{\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ (solid), and $k_0=0.05{\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ (dotted).