K-mouflage effects on clusters of galaxies

We investigate the effects of a K-mouflage modification of gravity on the dynamics of clusters of galaxies. We extend the description of K-mouflage to situations where the scalar field responsible for the modification of gravity is coupled to a perfect fluid with pressure. We describe the coupled system at both the background cosmology and cosmological perturbations levels, focusing on cases where the pressure emanates from small-scale nonlinear physics. We derive these properties in both the Einstein and Jordan frames, as these two frames already differ by a few percents at the background level for K-mouflage scenarios, and next compute cluster properties in the Jordan frame that is better suited to these observations. Galaxy clusters are not screened by the K-mouflage mechanism and therefore feel the modification of gravity in a maximal way. This implies that the halo mass function deviates from $\mathrm{\Lambda }$-CDM by a factor of order 1 for masses $M\gtrsim {10}^{14}{h}^{-1}{M}_{\odot }$. We then consider the hydrostatic equilibrium of gases embedded in galaxy clusters and the consequences of K-mouflage on the x-ray cluster luminosity, the gas temperature, and the Sunyaev–Zel’dovich effect. We find that the cluster temperature function, and more generally number counts, are largely affected by K-mouflage, mainly due to the increased cluster abundance in these models. Other scaling relations such as the mass-temperature and the temperature-luminosity relations are only modified at the percent level due to the constraints on K-mouflage from local Solar System tests.

Figure 1

Evolution with redshift of the matter and dark-energy cosmological parameters ${\mathrm{\Omega }}_{\mathrm{m}}(z)$ and ${\mathrm{\Omega }}_{\text{de}}(z)$. We display the two K-mouflage models of Eqs. (71) (arctan model, red crosses) and (74) (cubic model, blue squares) and the reference $\mathrm{\Lambda }$-CDM universe (black dashed lines). The two scalar-field models almost coincide in this figure.

Figure 2

Relative deviation of the Hubble expansion rate with respect to the $\mathrm{\Lambda }$-CDM reference, $\mathrm{\Delta }H/H=H/H_{\mathrm{\Lambda }\text{−}\text{CDM}}-1$, for the same K-mouflage models as in Fig. 1.

Figure 3

Effective equation of state parameters $w_{\text{de}}(z)$ (solid lines with a divergence and change of sign at $z\simeq 3$) and $w_{\phi }(z)$ (dashed lines with a smooth behavior), for the same K-mouflage models as in Fig. 1.

Figure 4

Upper panel: background scalar field $\overline{\phi }$ as a function of redshift. Lower panel: background kinetic term $\overline{\tilde{\chi }}$ as a function of redshift.

Figure 5

Coefficients ${\epsilon }_1$ and ${\epsilon }_2$, defined in Eqs. (46) and (25) for the K-mouflage models, as functions of redshift.

Figure 6

Relative drift with redshift of the effective Newton constant for the K-mouflage models.

Figure 7

Relative deviation, $D_{+}/D_{+\mathrm{\Lambda }\text{−}\mathrm{CDM}}-1$, of the linear growing mode $D_{+}$ from the $\mathrm{\Lambda }$-CDM reference.

Figure 8

Relative deviation, $f/f_{\mathrm{\Lambda }\mathrm{CDM}}-1$, of the linear growth rate, $f(z)=\mathrm{d}\ln D_{+}/\mathrm{d}\ln a$, from the $\mathrm{\Lambda }$-CDM reference.

Figure 9

Linear density contrast threshold ${\delta }_{L(\mathrm{\Lambda })}$ associated with the nonlinear density contrast ${\mathrm{\Delta }}_c=200$ with respect to the critical density ${\rho }_{\text{crit}}$, for the K-mouflage models and the $\mathrm{\Lambda }$-CDM reference.

Figure 10

Relative deviation, $n(M)/n_{\mathrm{\Lambda }\text{−}\mathrm{CDM}}(M)-1$, of the halo mass function of the K-mouflage models from the $\mathrm{\Lambda }$-CDM reference, at $z=0$ (solid lines) and $z=2$ (dashed lines). Halos are defined by the density contrast ${\mathrm{\Delta }}_c=200$ with respect to the critical density.

Figure 11

Upper panel: relative deviation of the matter density power spectrum from the $\mathrm{\Lambda }$-CDM reference, at $z=0$ (solid lines) and $z=2$ (dashed lines), for the K-mouflage models. Lower panel: relative deviation of the matter density correlation function from the $\mathrm{\Lambda }$-CDM reference.

Figure 12

Mass-concentration relation for NFW halos, for the K-mouflage models and the $\mathrm{\Lambda }$-CDM reference, at $z=0.37$. The black points (with their error bars) are observational measures taken from Ref. [31].

Figure 13

Upper panel: scalar-field radial profile, $\delta \phi (r)=\phi (r)-\overline{\phi }$, for halos of mass ${10}^{15}{h}^{-1}{M}_{\odot }$ (upper solid lines) and ${10}^{13}{h}^{-1}{M}_{\odot }$ (lower dashed lines). The scalar-field fluctuation is negative as the scalar field $\phi$ is minimum at the center of the halo. Lower panel: radial profile of the “kinetic energy” $-\tilde{\chi }(r)$ of the scalar field, for halos of mass ${10}^{15}{h}^{-1}{M}_{\odot }$ (upper solid lines) and ${10}^{13}{h}^{-1}{M}_{\odot }$ (lower dashed lines). Here $\tilde{\chi }<0$ because we consider the static limit, which is dominated by spatial gradients. The arctan and cubic K-mouflage models give almost identical results in this figure.

Figure 14

Upper panel: density profiles for a cluster of mass $M_{200c}=5\times {10}^{14}{h}^{-1}{M}_{\odot }$. The upper solid lines refer to the dark matter density profiles and the lower dotted lines to the gas density profiles. The K-mouflage models and the $\mathrm{\Lambda }$-CDM reference cannot be distinguished in this figure. Lower panel: relative deviation from the $\mathrm{\Lambda }$-CDM reference of the dark matter (solid lines) and gas (dotted lines) density profiles.

Figure 15

Upper panel: mass-temperature relation for the K-mouflage models and the $\mathrm{\Lambda }$-CDM reference, at $z=0.0288$ (lower curves) and $z=0.451$ (upper curves). The data points are taken from observations made by Refs. [36] (in green), [37] (in magenta), and [38] (in brown), with clusters in the redshift range $0.0288\le z\le 0.451$. Lower panel: relative deviation of the cluster mass-temperature relation from the $\mathrm{\Lambda }$-CDM reference, at $z=0.0288$.

Figure 16

Upper panel: temperature-luminosity relation for the K-mouflage models and the $\mathrm{\Lambda }$-CDM reference, at $z=0.048$ (lower curves) and $z=0.451$ (upper curves). The data points are taken from observations made by Refs. [39] (in green) and [38] (in brown), with clusters in the redshift range $0.048\le z\le 0.451$. Lower panel: relative deviation of the cluster temperature-luminosity relation from the $\mathrm{\Lambda }$-CDM reference, at $z=0.048$.

Figure 17

Upper panel: cluster temperature function for the K-mouflage models and the $\mathrm{\Lambda }$-CDM reference, at $z=0.05$. The data points are taken from observations made by Ref. [40] from a sample of clusters with $z\simeq 0.05$. Lower panel: relative deviation of the cluster temperature function from the $\mathrm{\Lambda }$-CDM reference at $z=0.05$.

Figure 18

Integrated Comptonization within $R_{500c}$ as a function of the gas mass (upper panel) and gas temperature (lower panel) for the K-mouflage models and the $\mathrm{\Lambda }$-CDM reference. These different models cannot be distinguished in these figures. We show our results for $z=0.16$ (lower curves in the upper panel and upper curves in the lower panel) and $z=1.45$ (upper curves in the upper panel and lower curves in the lower panel). The data points are measures from a sample of clusters in the range $0.16\le z\le 1.45$ [41].

Figure 19

Relative deviation of the cluster correlation function from the $\mathrm{\Lambda }$-CDM reference for the K-mouflage models. We consider halos of mass ${10}^{15}{h}^{-1}{M}_{\odot }$ (upper solid lines) and ${10}^{13}{h}^{-1}{M}_{\odot }$ (lower dashed lines).

Figure 20

Relative deviation, $\tilde{H}/H-1$, of the Einstein-frame Hubble rate from the Jordan-frame Hubble rate, as a function of the Jordan-frame redshift.

Figure 21

Deviation of the Einstein-frame cosmological parameters from their Jordan-frame counterparts, $\widehat{\mathrm{\Omega }}-\mathrm{\Omega }$, as a function of the Jordan-frame redshift. We show the matter density parameters (solid lines) and the dark-energy density parameters (dashed lines). In the Einstein frame, we consider the effective matter and dark-energy densities as given by Eqs. (b2) and (b3).