From the light-cone simulations of the different cosmological models, we can com-
pute the moments of they and b distortion for the thermal and kinetic maps. In
Table 3.2, we report the values obtained averaging over 8 realizations of 3 degrees
square map for each cosmology. We can see that all theymean values are well below
the observational constraints reported by the COBE–FIRAS experiment, which sets
a 95 per cent upper limit ofymean <1.5×10−5(Fixsen et al., 1996). In the ΛCDM
model, we obtain a value of hyi = 1.55×10−6 as the mean of the pixel values in
the thermal maps and hbi = 1.78×10−7 for the kinetic ones. These values are
slightly lower than the result of 3.2×10−6 obtained by da Silva et al. (2000) and
the 2.5×10−6 found by White et al. (2002) for pure non-radiative runs, whereas,
more recently, Roncarelli et al. (2007) found 1.19×10−6when analyzing simulations
that included cooling, star formation and feedback. The main reasons of this dis-
crepancy is the smaller value ofσ8adopted in our simulations (σ8= 0.8) compared
to the first two studies (σ8 = 0.9), and the inclusion of extra physics in the third
one. The Comptony-parameter scales roughly withσ8α/2, withα≈4−7 (see, e.g.,
Sadeh and Rephaeli, 2004; Diego and Majumdar, 2004). Also, the inclusion of ad- ditional physics can affect the results, in particular cooling reduces the contribution
of high density gas in groups and clusters, loweringymean by about 20% .
The mean Comptonization we predict for the EDE model is systematically higher than the one expected in the ΛCDM cosmology. For the EDE1 model we find
ymean ∼ 1.79×10−6, which is 15% higher with respect to a standard model with
the same cosmological parameters. We note however that the lowest and highest values for the distortion are seen in the dark energy model EDE2P and EDE3P, that also adopt a different cosmology. There is almost a factor two difference between
these two runs. In fact, the meany and b parameters are quite sensitive to theσ8
normalization and to the different Hubble expansion. Both of these cosmologies use a lower power spectrum normalization today, and this effect is dominant in EDE3P, which has the lowest values for the mean, but not in the model EDE2P, since in this model accelerated Hubble expansion dominates. These two cosmologies also show systematically lower skewness and kurtosis. The relative order of the models is exactly the same when we consider the intensity of the kinetic SZ signals instead. Here the increase in the EDE models reflects the impact of the dark energy on the expansion history of the Universe (see Figure 3.3).
In Figure 3.10, we plot the probability distribution of pixel values for both the thermal and kinetic SZ effects. We show as separate curves the results averaged over 8 different light-cone realizations for different cosmological models. The distribution
of logy is nearly symmetric and close to a log-normal function, reflecting the non-
Gaussianity of the y parameter (Seljak et al., 2001). Based on the values of the
skewness of the distribution we found out that the non-Gaussian effect is almost the same in all cosmologies, and shows only a small shift towards high Comptonization
region in the EDE models.
On the other hand, the probability distribution function for theb parameter in
the ΛCDM cosmology is well approximated by a Gaussian, despite the occurrence of rare bright events. A distinctive feature of the dark energy models is a strong depression in the peak of the distribution, and a significant increase of the kurtosis, thus the tails of distribution are more populated. In fact, the kSZ effect has a non-
negligible contribution on small-scales coming from high-z clusters, and then the
signal from non-linear structures is partly canceled out. However, when considering the distribution of the signal in different redshift slices, we recover the Gaussian shape in all the cosmologies, see Figure 3.11. Since in a given redshift bin there are gas elements both approaching and receding from the observer with equal probabil- ity, the kSZ signal has a vanishing expectation value. The two cosmologies of the EDE2P and EDE3P again delimit the upper and lower limits of the pixels values that are obtained. In general, we observe more extended tails in models that show the higher mean Comptonization parameter, and we conclude that the probability to obtain high values of the kinetic signal is enhanced by the accelerated expansion of the Universe in the EDE models.
In Figure 3.12 we plot the differential and integrated contributions as a function
of redshifts for the meany-distortion in all the different cosmologies. These values
represent the average over 8 light-cone realizations, and are computed in equally
spaced comoving distance intervals of length 100h−1Mpc out to redshift 10. The
upper panel shows the peak of the meany-distortion as a function of redshift. One
can notice that there is a large scatter at lower redshift (z <1), mainly due to the
probability of finding a very bright cluster in these particular redshift bins. The spikes disappear at higher redshift, since the light-cones include larger comoving volumes at larger distances and then the more massive clusters contribute a smaller
fraction of the total meany-Comptonization. Moreover, large collapsed structures
are very rare at higher redshift.
From this plot, we cannot really tell the different models apart, while we can easily trace the differences looking at the integrated distribution of the mean Comp- tonization, in the lower panel of Figure 3.12. The area under the curves quantifies the cumulative mean distortion at that time. We can notice that close to redshift zero all the cosmologies behave in a similar way, given that they were normalized
to the same σ8 today and they reproduce the same cluster temperature function
today. At higher redshift the growth factor evolution is slower for the models with early dark energy, and this anticipates the formation of the structures with respect to a ΛCDM model with the same cosmological parameters. The cumulative effect of the increased hot gas abundance gives rise to a mean thermal distortion sys- tematically larger in the maps that refer to EDE cosmologies. For example, in the ΛCDM case, most of the signal comes from redshifts less than one, and only about
da Silva et al., 2000). In contrast, in the EDE cases the tail extends to much higher
redshift, because structure grows there already. What is remarkable is that atz >2
the contribution to the tSZ effect is non-negligible in the EDE cases. Finally, we ob- serve that at very small redshift, it is the ΛCDM cosmology which gives the greatest signal, even though structures form earlier in the EDE cases. The reason for this is that the ΛCDM cosmology has the greatest volume element at these redshifts, and so a larger amount of gas contributes to the backwards light-cone.
Figure 3.13 shows the differential redshift distribution of σkSZ (upper panel),
averaged over the 8 maps, for the same cosmology analyzed before. The variance provides the complete description of the signal for a Gaussian distribution of the pixel values, therefore the curves give a measurement of its dispersion. We see again that the variance of the SZ signal in all cosmologies comes from a broad range of redshifts out to around 2, and falls off significantly only beyond that. The signal from nearby redshifts is primarily due to clusters with very high peculiar velocities, while at higher redshifts the number of rare events is smaller and the distribution is narrower. In fact, although we averaged over 8 light-cone realizations, we have only one simulation per model, which is not sufficient to completely eliminate the cosmic variance. Integrating the redshift distribution over the maps (lower panel) we see again that we have a more significant contribution from high-redshift sources in the EDE models compared with the ΛCDM model. Unlike for the integrated
thermal effect, the gas mass atz >6 still adds important contributions to the total
kinetic SZ effect. The distribution is in fact not convergent up to very high redshift. However, the peak of the thermal effect gives us additional information where the strongest kinetic signals in the maps are expected. Finally, the gray dashed line in the plot refers to the model EDE3P. We notice that the integrated variance is
reduced by 30% already at redshiftz= 3. This result confirms the expectation that
the kinetic effect is a sensitive function ofσ8.