Multiwavelength constraints on the inflationary consistency relation

We present the first attempt to use a combination of CMB, LIGO, and Pulsar Timing Array (PPTA) data to constrain both the tilt and the running of primordial tensor power spectrum through constraints on the gravitational wave energy density generated in the early universe. Combining measurements at different cosmological scales highlights how complementary data can be used to test the predictions of early universe models including the inflationary consistency relation. Current data prefer a slightly positive tilt ($n_t=0.06_{-0.89}^{+0.63}$) and a negative running ($n_{t,\text{run}}<-0.22$) for the tensor power spectrum spectrum. Interestingly, the addition of direct gravitational wave detector data alone puts strong bounds on the tensor-to-scalar ratio $r<0.2$ since the large positive tensor tilt preferred by the Planck temperature power spectrum is no longer allowed. Adding the recently released BICEP2/KECK and Planck 353 GHz polarization cross-correlation data gives an even stronger bound $r<0.1$. We comment on possible effects of a large positive tilt on the background expansion and show that depending on the assumptions regarding the UV cutoff ($k_{\mathrm{UV}}/k_{*}=10^{24}$) of the primordial spectrum of gravitational waves, the strongest bounds on $n_t=0.07_{-0.80}^{+0.52}$ are derived from this effect.

Figure 1

The marginalized 2D posterior constraints on $n_t$ and $r$. The large grey contour shows the constraints of using BICEP2 in conjunction with CMB data from Planck [34], the HST prior on the Hubble constant [35], and the constraints on the acoustic scale from BAO [36] measurements (the latter three combined in the name “All”). Adding in data from the PPTA constraints on the gravitational wave energy density improves the constraints threefold on $n_t$, while the constraint on $r$ is largely unchanged. Using data from LIGO instead tightens this even further, highlighting the constraining power of LIGO on the primordial tensor power spectrum. All contours agree on the value of $r$ announced by the BICEP2 team, confirming that the LIGO and PPTA data are not particularly sensitive to the value of $r$ directly, but they are sensitive to the tilt $n_t$, as shown in Eq. (3).

Figure 2

The marginalized contours in the $r$–$n_t$ plane comparing BICEP2 to the BKP data. If all the signal in BICEP2 is primordial, we obtain strong bounds on $n_t$, with a $2\sigma$ preference for $n_t>0$. The upper limit is driven by the LIGO constraint. From the BICEP2/KECK cross Planck analysis $r$ is lowered and therefore in combination with the constraints from Planck $TT$ power spectrum leads to a shift in the peak value of $n_t$. With a small value of $r$, constraints on $n_t$ are weakened. However, including LIGO puts a strong upper bound on $n_t$ as before.

Figure 3

The marginalized contours in the $r$–$n_t$ plane without using BICEP2 data. When $r_{0.01}=0$ there is no constraint on $n_t$ in principle, however Planck $TT$ can be best fit with $r_{0.01}=0.24$, hence there exists some constraint on $n_t$. Planck prefers a very blue tensor spectrum, in order to create an artificial running of the temperature data on large scales. From this analysis it is also apparent that Planck drives the value of $n_t$, not BICEP2 data. Interestingly, by adding LIGO, such large values of $n_t$ are ruled out, which has the effect that $r$ is significantly constrained $r_{0.01}<0.2$ at 95%.

Figure 4

The contribution from the tensors to the $TT$ power spectrum. A very large positive tilt leads to a rescaling of the spectrum, except on very large scales, where the spectrum is suppressed. This suppression is what Planck prefers (though not significantly) over no tensor modes at all.

Figure 5

Constraints allowing for more freedom in the GW power spectrum. Left panel: The marginalized 2D constraints in the $r$–$n_t$ plane for the cases including BICEP2 (bottom) and BICEP2/KECK cross Planck (top) data. Right panel: Constraints on the $n_t-n_{t,\text{run}}$ for the same cases. Allowing for running in the tensor spectral index does not change the constraint on the tilt significantly, as the errors on the running are large and there is no significant correlation between the two parameters. BICEP2 data improve constraints on $n_t$ as expected. The addition of $n_{t,\text{run}}$ generally weakens the constraint on $r$. Note the different ticks in the figures on the right.

Figure 6

Left: The relative change in the scale of matter-radiation equality as a function of the tensor tilt $n_t$, assuming $r=0.2$. The simple analytical estimate suggests that $n_t\sim 0.4$ would be modifying the scale of equality at the 5% level, which is close to current bounds from WMAP and Planck. Obviously, for smaller $r$, this constraint weakens accordingly. Right: The change in $N_{\text{eff}}$ as a function of $n_t$ for $r=0.2$ using the two approximations given in Eqs. (11) and (12). The difference is very small up until the energy density in gravitational waves becomes of the order of the total energy density in the Universe. Our constraints are derived using Eq. (11). A detailed explanation of both curves and how these are derived can be found in the text.

Figure 7

The constraints on the gravitational wave spectrum when adding in the change in $N_{\text{eff}}$ from gravitational waves. Allowing for the $N_{\text{eff}}$ contribution cuts off the distribution on $n_t$ to more positive values. Adding in BICEP2 data again moves the constraints to more positive values of $r$, as expected. When considering the BKxP data in addition to Planck, the upper limits on $n_t$ are hardly affected as can been see in the figure on the right. We also show the effect of changing the cutoff scale from $k_{*}/k_{\mathrm{UV}}=10^{20}$ to $k_{*}/k_{\mathrm{UV}}=10^{24}$. The overall effect is to limit the upper bound on $n_t$.

Figure 8

A scatter plot showing the relation between the tensor tilt $n_t$, the Hubble rate $H_0$ and the redshift of equality $z_{\text{eq}}$, including the contribution from gravitational waves to $N_{\text{eff}}$. The top panel shows the case where the data are combined with BICEP2, while the middle and lower panels replace the BICEP2 data with the BICEP2/KECK-Planck cross spectrum for two different values of the cutoff. In all cases, $z_{\text{eq}}$ is fairly well constrained, hence in order to accommodate large $n_t$ (and hence a large $N_{\text{eff}}$, $H_0$ is increased by adding more matter to the Universe. Because of the steep increase of $N_{\text{eff}}$ as a function of $n_t$ there is a very strong turn in degeneracy between $n_t$ and $H_0$. There is a shift in the upper bound of $n_t$ allowed by BICEP2 compared to the cross spectrum, and this limit also shifts with different values of the cutoff, as can be seen by comparing the middle to lower panels of the figure.