First observational tests of eternal inflation: Analysis methods and WMAP 7-year results

In the picture of eternal inflation, our observable universe resides inside a single bubble nucleated from an inflating false vacuum. Many of the theories giving rise to eternal inflation predict that we have causal access to collisions with other bubble universes, providing an opportunity to confront these theories with observation. We present the results from the first observational search for the effects of bubble collisions, using cosmic microwave background data from the WMAP satellite. Our search targets a generic set of properties associated with a bubble-collision spacetime, which we describe in detail. We use a modular algorithm that is designed to avoid a posteriori selection effects, automatically picking out the most promising signals, performing a search for causal boundaries, and conducting a full Bayesian parameter estimation and model selection analysis. We outline each component of this algorithm, describing its response to simulated CMB skies with and without bubble collisions. Comparing the results for simulated bubble collisions to the results from an analysis of the WMAP 7-year data, we rule out bubble collisions over a range of parameter space. Our model selection results based on WMAP 7-year data do not warrant augmenting $\Lambda \mathrm{CDM}$ with bubble collisions. Data from the Planck satellite can be used to more definitively test the bubble-collision hypothesis.

Figure 1

The radial temperature modulation Eq. (4) induced by a bubble collision centered on the north pole ($\theta =0$).

Figure 2

A Poincaré-disk representation of the surface of last scattering inside our parent bubble, with one dimension suppressed. The future light cone of the collision at this time is denoted by the boundary of the shaded region; the latter represents the portions of the surface of last scattering that are to the future of the collision. Our past light cone at last scattering is represented by the dashed circle. From the present bounds on curvature, the size of our past light cone must be much smaller than 1 curvature radius. Zooming in on the portion of the surface of last scattering that we have causal access to (inset), the Universe is very close to flat, and the region affected by the collision has approximate planar symmetry. The region affected by the collision appears as a disk of angular radius ${\theta }_{\mathrm{crit}}$ on the CMB sky.

Figure 3

On the left, we show a bubble-collision template with $\{z_0=5.0\times {10}^{-5},z_{\mathrm{crit}}=-5.0\times {10}^{-5},{\theta }_{\mathrm{crit}}=10.0°,{\theta }_0=57.7°,{\varphi }_0=99.2°\}$. On the right we add simulated background fluctuations, smoothing, and instrumental noise.

Figure 4

The two locations chosen for our simulated bubble collisions are overplotted on the WMAP 7-year noise variances with the KQ75 7-year mask applied. The regions encompassed by the 5° and 10° simulated collisions are shaded black and gray, respectively. Bubbles are centered in an unmasked high-noise $\{{\theta }_0=56.6°,{\varphi }_0=193.0°\}$ (left) and low-noise $\{{\theta }_0=57.7°,{\varphi }_0=99.2°\}$ region (right).

Figure 5

The filter function $b(\ell ,B,j)$ for standard (left panel) and Mexican (right panel) needlets with $B=2.5$ for $j=0$, 1, 2, 3 (solid, dashed, dot-dashed, dotted curves).

Figure 6

Standard needlets in pixel space. On the left, we show standard needlets ${\psi }_{jk}$ with $B=2.5$ for $j=1$, 2, 3 (dot-dashed, dashed, solid curves) at fixed $k$ as a function of the polar angle $\theta$. On the right, we show standard needlets ${\psi }_{jk}$ for fixed $j=3$ at three pixels $k$ (dashed, solid, dot-dashed curves) as a function of the polar angle $\theta$ (note: since we are projecting onto $\theta$, the needlets appear asymmetric).

Figure 7

The spherical harmonic transform (connected dots) of a ${\theta }_{\mathrm{crit}}=5°$ (left panel) and ${\theta }_{\mathrm{crit}}=25°$ (right panel) bare collision template centered on the north pole on top of the filter function $b(\ell ,B,j)$ for standard needlets with $B=2.5$ for $j=0$, 1, 2, 3 (solid, dashed, dot-dashed, dotted curves). The overlap of the filter function $b(\ell ,B,j)$ with the spherical harmonic transform of the bubble template [see Eq. (21)] determines for which value of $j$ the needlet transform yields the largest signal. In these examples, the 5° collision has the largest needlet response at $j=3$ and the 25° collision at $j=2$. The needlet response as a function of angle for the 25° collision is plotted in Fig. 8.

Figure 8

$B=2.5$ for standard needlets and a 25° collision. $j=0$ (circles), $j=1$ (squares), $j=2$ (diamonds), and $j=3$ (triangles).

Figure 9

Needlet-coefficient variance maps for standard needlets with $B=2.5$, $j=2$ (top row) and Mexican needlets with $B=1.4$, $j=11$ (bottom row), generated using an ensemble of 3000 Gaussian CMB realizations. The map-averaged variances of the full-sky maps (left column) agree well with expectations from theory [Eq. (23) predicts $774\mu {\mathrm{K}}^2$ and $150\mu {\mathrm{K}}^2$ respectively], as do regions of sky sufficiently distant from the 7-year KQ75 mask (right column). The low-frequency needlets are affected predominantly by the Galactic cut, whereas the high-frequency needlets are affected by point-source masking.

Figure 10

The temperature (top panels) and needlet-coefficient significance (bottom panels) maps for a simulated bubble collision with $z_0=5\times {10}^{-5}$, $z_{\mathrm{crit}}=-5\times {10}^{-5}$, ${\theta }_{\mathrm{crit}}=10°$ on the CMB sky with the WMAP 7-year KQ75 mask applied. We show the map of needlet coefficients which gives the largest needlet response, in this case $j=3$ for standard needlets with $B=2.5$. The right-hand panels provide close-ups of the collision region.

Figure 11

Exclusion (black) and sensitivity (gray) regions for the needlet step of the analysis pipeline applied to a set of 5° (left panel) and 10° (right panel) simulated bubble collisions. If all realizations at the high- and low-noise locations yield a detection, we include the collision in the exclusion region; such collisions would certainly be detected as long as they were not significantly masked. If a detection is made in any realization/location, we include the collision in the sensitivity region; such collisions could be found if they were in a favorable location of the sky (i.e., low noise, or a region with a specific realization of background CMB fluctuations).

Figure 12

An illustration of the Canny algorithm for edge detection. The input temperature map contains a circular discontinuity, which can be seen in a map of the gradient magnitude as a local maximum. Nonmaximal suppression selects for local maxima in the gradient map. The hysteresis thresholding step finds stitched edges by comparing the local direction of the gradient at adjacent pixels.

Figure 13

A depiction of the circular Hough transform (CHT). On the left is a Boolean map of edge pixels, as output by the Canny algorithm. Centering a pair of arcs oriented in the direction of the local gradient about each edge pixel, the CHT counts the number of times each pixel is intersected. The presence of a circular edge is indicated by a maximum in the CHT score—the hit count divided by the arc radius—as the arc radii are varied. On the left, we show a set of arcs, centered on four pixels on the circular edge we wish to detect; there will be no clear peak in the CHT score for this radius. Increasing the arc radius to match that of the circular edge (center), there will be a large number of hits at the true center of the edge. On the right, we show the actual map of the CHT score at each pixel for this example. As this has been scanned at the correct angular scale, there is a large peak at the center of the circular edge.

Figure 14

The temperature map (left panel) and CHT score (right panel) for our illustrative example of a 10° bubble-collision simulation. The CHT score is recorded at each pixel passing the needlet significance threshold. For a search radius of 10° there is a clear peak in the CHT score at the center of the simulated collision.

Figure 15

The global maximum of the CHT score at each circle radius for the collision simulation shown in Fig. 14. The collision has a maximum response for standard needlets with $B=2.5$ at $j=3$, from which Table  sets the search range to be $5°\le {\theta }_{\mathrm{crit}}\le 14°$. The peak of the CHT score at 10° clearly identifies the correct angular scale of the simulated collision.

Figure 16

The most edgelike feature in the WMAP end-to-end simulation. The contrast in scores as a function of position (left panel) and radius (right panel) is greatly reduced compared to the collision example (Figs. 14, 15).

Figure 17

Exclusion (black) and sensitivity (gray) regions (see Fig. 11) for the edge-detection step of the analysis pipeline applied to a set of 5° (left panel) and 10° (right panel) simulated bubble collisions.

Figure 18

The normalized posterior $\mathrm{Pr}({\overline{N}}_s|N_b,f_{\mathrm{sky}})$ [see Eq. (a16)] assuming $f_{\mathrm{sky}}=0.7$. In the left panel, we show the posterior obtained for one blob $N_b=1$ for a local evidence ratio ${\rho }_b=4$. Comparing with the posterior at ${\overline{N}}_s=0$ (dashed line), we see that any theory predicting ${\overline{N}}_s\lesssim 4$ will be preferred over the theory without bubble collisions. In the right panel, we show the posterior obtained for four blobs with identical local evidence ratios ${\rho }_b=1/2$. Again, comparing with the posterior at ${\overline{N}}_s=0$, any theory with ${\overline{N}}_s\lesssim 7$ will be preferred over the theory without bubble collisions. When there are multiple blobs, the bubble-collision hypothesis can be supported even when the evidence ratio for each blob is less than 1.

Figure 19

The full posterior $\mathrm{Pr}({\overline{N}}_s|N_b,f_{\mathrm{sky}})$ [Eq. (a16)] that would be obtained from a conclusive detection (i.e., ${\rho }_{b_s}\gg 1$) of $N_b=0$, 2, 4 (solid, dashed, and dot-dashed curves) blobs containing bubble collisions assuming $f_{\mathrm{sky}}=0.7$. The presence of a sky cut skews the distribution toward ${\overline{N}}_s>N_b$. Note that even when features are conclusively detected, there is an intrinsic uncertainty in ${\overline{N}}_s$; this is a form of cosmic variance.

Figure 20

The clearest peak found during the edge-detection analysis of the WMAP data. The contrasts in scores as a function of position (left panel) and radius (right panel) are comparable to those obtained in the analysis of the end-to-end simulation (Fig. 16), and greatly reduced compared to the collision example (Figs. 14, 15).

Figure 21

Full-sky map showing the positions and sizes of the four features with largest evidence ratios, along side the 7-year KQ75 sky cut. The features are plotted clockwise from the blob closest to the Galactic mask: feature 3 (red), feature 2 (orange), feature 7 (light blue), and feature 10 (light green).

Figure 22

Maps of the four features with largest evidence ratios. The top row shows the W-band temperature map in the locality of the four features, masked with the KQ75 mask. Overlaid are circles indicating the estimated position and angular scale found in each case. The second row contains plots of the masked needlet significances for the needlets whose ${\theta }_{\mathrm{crit}}$ priors produced the largest evidence ratios. These plots appear pixelated as the blob-detection step is carried out at reduced resolution. The third row shows the bubble-collision templates corresponding to the estimated model parameters; these templates are subtracted from the W-band data in the fourth row. The width of each plot is $\sim 16.7°$.

Figure 23

WMAP channel frequency dependence of the features highlighted by our pipeline. The W-band-normalized difference in pixel-averaged needlet coefficients between the Q and W bands (triangles) and V and W bands (diamonds) are plotted for each blob making up a feature, as highlighted by the full suite of needlet transforms.