Robust analysis of the growth of structure

Current cosmological tensions show that it is crucial to test the predictions from the canonical $\mathrm{\Lambda }\mathrm{CDM}$ paradigm at different cosmic times. One very appealing test of structure formation in the Universe is the growth rate of structure in our universe $f$, usually parametrized via the growth index $\gamma$, with $f\equiv {\mathrm{\Omega }}_m(a{)}^{\gamma }$ and $\gamma \simeq 0.55$ in the standard $\mathrm{\Lambda }\mathrm{CDM}$ case. Recent studies have claimed a suppression of the growth of structure from a variety of cosmological observations, characterized by $\gamma >0.55$. By employing different self-consistent growth parametrizations schemes, we show here that $\gamma <0.55$, obtaining instead an enhanced growth of structure today. This preference reaches the $3\sigma$ significance using cosmic microwave background observations, supernova Ia and baryon acoustic oscillation measurements. The addition of cosmic microwave background lensing data relaxes such a preference to the $2\sigma$ level, since a larger lensing effect can always be compensated with a smaller structure growth, or, equivalently, with $\gamma >0.55$. We have also included the lensing amplitude $A_{\mathrm{L}}$ as a free parameter in our data analysis, showing that the preference for $A_{\mathrm{L}}>1$ still remains, except for some particular parametrizations when lensing observations are included. We also do not find any significant preference for an oscillatory dependence of $A_{\mathrm{L}}$, $A_{\mathrm{L}}+A_m\sin \ell$. To further reassess the effects of a nonstandard growth, we have computed by means of $N$-body simulations the dark matter density fields, the dark matter halo mass functions and the halo density profiles for different values of $\gamma$. Future observations from the Square Kilometer Array, reducing by a factor of 3 the current errors on the $\gamma$ parameter, further confirm or refute with a strong statistical significance the deviation of the growth index from its standard value.

Figure 1

Left panel: time dependence of the parameter $\mu (a)$ together with the 1 and $2\sigma$ errors from CMB temperature and polarization plus CMB lensing measurements. Right panel: equivalence between the $\gamma$ parameter and $\mu (z)-1$; see Eq. (9), for a number of different redshifts. The cross point corresponds to $\mathrm{\Lambda }\mathrm{CDM}$ and the horizontal and vertical dashed lines depict the expectations within standard cosmology, i.e., $\mu =1$ and $\gamma =0.55$ respectively.

Figure 2

Left panel: 68% and 95% CL contours in the (${\eta }_0-1$, ${\mu }_0-1$) plane within the growth model parametrized via $\mu$ and $\eta$, see Eq. (10), obtained by using CMB temperature and polarization measurements (blue contours) and CMB temperature, polarization and lensing observations (red contours). Right panel: as in the left panel but when the lensing amplitude $A_{\mathrm{L}}$ is also considered as a freely varying parameter.

Figure 3

Left panel: 68% and 95% CL contours in the (${\eta }_0-1$, $\gamma$) plane within the growth model parametrized via $\gamma$ and $\eta$; see Eq. (9) and the second of Eq. (10) obtained by using CMB temperature and polarization measurements (blue contours) and CMB temperature, polarization and lensing observations (red contours). Right panel: as in the left panel but when the lensing amplitude $A_{\mathrm{L}}$ is also considered as a freely varying parameter.

Figure 4

Left panel: the red regions depict the 68% and 95% CL contours in the (${\eta }_0-1$, ${\mu }_0-1$) plane within the growth model parametrized via $\mu$ and $\eta$; see Eq. (10) obtained by using CMB temperature and polarization measurements, BAO, SNIa, weak lensing and RSD measurements. The blue regions depict the equivalent but when the lensing amplitude $A_{\mathrm{L}}$ is also considered as a freely varying parameter. Right panel: allowed regions in the simplest growth parametrization used here, i.e., via just one single parameter $\gamma$ in the ($\gamma$, $A_{\mathrm{L}}$) plane allowed from different datasets, for the sake of comparison with the results of Ref. [26]. Thee horizontal and vertical lines depict the expectations within the canonical $\mathrm{\Lambda }\mathrm{CDM}$ cosmology, i.e., $A_{\mathrm{L}}=1$ and $\gamma =0.55$, respectively.

Figure 5

Left panel: CMB lensed power spectrum, together with Planck 2018 measurements for several possible scenarios; see text for details. Right panel: deviation of the lensing potential with respect to the $\mathrm{\Lambda }\mathrm{CDM}$ case for different growth parametrizations.

Figure 6

$S_0$ acts as a signal parameter to measure the deviation from the standard cosmic growth in general relativity. The red and blue points denote the best fit value and $2\sigma$ limits of $S_0$, respectively. Similarly, the red and blue lines are the best fit line and $2\sigma$ boundaries when using the fitting formula $S_0={\mu }_0+0.4{\eta }_0$, respectively. The magenta dashed line corresponds to ${\eta }_0=0$, and the cross point between gray dashed lines represents general relativity. The blue contours depict the 68% and 95% confidence regions in the (${\eta }_0$, ${\mu }_0$) plane within the growth model parametrized via $\mu$ and $\eta$, see Eq. (10), obtained by using CMB temperature, polarization and lensing observations.

Figure 7

Density fields of dark matter in the growth index $\gamma$ model for the values $\gamma =0.506$, 0.3 and 0.7 at redshift $z=0$. For the case of $\gamma =0.506$, we use the best fit of the growth index model to implement the simulation. We also depict the $\mathrm{\Lambda }\mathrm{CDM}$ case as a reference and choose a slice of $20h^{-1}\mathrm{Mpc}$ (i.e., $200\times 200\times 20h^{-3}{\mathrm{Mpc}}^3$) density field and stack it along $Z$ axis for each model. $\delta$ denotes the overdensity of dark matter.

Figure 8

Left panel: the dark matter halo mass functions in the growth index model are, respectively, shown for different values $\gamma =0.506$, 0.3 and 0.7 at redshift $z=0$. We also depict the halo mass function of the $\mathrm{\Lambda }\mathrm{CDM}$ case as a reference. The shaded regions denote the Poisson errors for each model. Right panel: the dark matter halo density profiles for different models are shown.