tokens: the halo contribution We estimate the average contribution from single LRG haloes as follows. Since we have selected central LRGs for our pair catalogue, we assume that the average single-halo contribution is circularly symmetric about each LRG in a pair. To determine the radial profile of each single halo, we analytially solve for the map (indexed by pixel p ) to a model of the form where y L,i is the single-halo signal in the ith radial bin centred on the ‘left’ LRG at (−1, 0), y R,j is the single-halo signal in the jth radial bin centred on the ‘right’ LRG at (+1, 0), and p = p ( X, Y ) is a pixel on the map as shown in Fig. 3 . When solving it, we choose radial bins of size Δ r = 0.02, and we weight the map pixels uniformly. To avoid biasing the profiles with non-circular filament signal, we exclude the central region −1 X +1 and −2 Y +2. Fig. 4 shows the resulting profiles for the LRG haloes. Figure 3. Illustration to estimate the radial profile of each single halo. The y value at a pixel p, y h ( p ), is the sum of ith radial bin centred on the left LRG at (−1, 0), y L, i ( p ), and jth radial bin centred on the right LRG, y R, j ( p ) at (+1, 0). To avoid biasing the profiles with non-circular filament signal, the central region −1 X +1 and −2 Y +2 is excluded. Figure 4. Top : The circular halo profiles estimated to the map in Fig. 2 . Bottom : The radial profile of the left and right haloes shown above. Fig. 5 shows the residual y map after subtracting the circular halo profiles shown in Fig. 4 (note the change in colour scale from Figs 4 and 5 ). The bright halo signals appear to be cleanly subtracted, while a residual signal between the LRGs is clearly visible. The lower panels of Fig. 5 show the residual signal in horizontal ( Y = 0) and vertical ( X = 0) slices through the map. The shape of the signal is consistent with an elongated filamentary structure connecting average pairs of central LRGs. The mean residual signal in the central region, −0.8 X +0.8 and −0.2 Y +0.2, is found to be Δ y = 1.31 × 10 −8 . Figure 5. Top : The residual y -map after the radial halo signals are subtracted. Bottom : The residual tSZ signal along the x- and y -axes. 3.3 Null tests and error estimates To assess the reality of the residual signal and estimate its uncertainty, we perform two types of Monte Carlo-based null tests. In the first test, we rotate the centre of each LRG pair by a random angle in galactic longitude, for example, the centre of one LRG pair is rotated from [ l, b ] = [10°, 60°] to [ l, b ] = [150°, 60°]. (Keeping the latitude fixed keeps the galactic foreground level in the y map approximately fixed.) We then stack the y map against the set of rotated LRG pairs, and we repeat this stacking of the full catalogue 1000 times to determine the rms fluctuations in the background (and foreground) sky. Fig. 6 shows one of the 1000 rotated, stacked y maps: the map has no discernible structure. We can use this ensemble of maps to
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