Universal pre-heating and the structure of the intergalactic gas
at the scales of groups and clusters of galaxies
G. González Casado1, J.M. Solanes2, A. Manrique2, E. Salvador Solé2
1Departament
de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, Pau Gargallo 5, Edificio U, E-08028 Barcelona, Spain
de Astronomia i Meteorologia, Universitat de Barcelona, Martí i Franquès 1, E-08028 Barcelona, Spain
2Departament
INTRODUCTION
We present a semi-analytic model for the evolution of the hot diffuse gas trapped within dark-matter halos that closely reproduces the X-ray observations at the scales of clusters and groups of galaxies. The model relays in a detailed and robust description of the formation,
evolution and origin of the internal structure of dark-matter halos which was developed in previous works and tested against N-body simulations. The evolution of the X-ray emitting gas in halos is described by means of a reduced set of physically and observationally
motivated assumptions which allow following the time evolution of the inner structure and global properties of the gas in halos of different masses, not only when the gas is simply shock-heated by gravitational interaction, but also when the gas has been preheated before
halo formation. In this poster we first compare our model predictions with the widely observed correlation between luminosity and temperature of the X-ray emitting gas at redshift zero. By means of that comparison we determine the best-fit values for the free parameters of
our model. Subsequently, we verify that such best-fitting values also provide a good description of the entropy, density and temperature structure of nearby groups and clusters in two and three dimensions.
OVERVIEW OF THE MODEL
DARK MATTER HALOS
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Hydrostatic equilibrium and spherical symmetry.
Mass distribution follows a NFW profile for any mass and time.
The time evolution of halo concentration and its mass dependence follow the empirical law derived by Eke et al. (2001, ApJ,
554) from the analysis of high resolution cosmological simulations.
The formation time distribution of halos is inferred from the semi-analytic model for halo evolution described in Raig, GonzálezCasado & Salvador-Solé (2001, MNRAS, 327). This work shows that the model predictions are consistent with N-body
simulation results.
Halo formation occurs at the last major merger and relaxation is instantaneous. Subsequently, halos grow inside-out accreting
mass at a rate described by the model mentioned in point 4. As shown by Manrique et al. (2003, ApJ, 593) the resulting
evolution closely reproduces the halo density profiles obtained in high resolution N-body simulations.
The model has no free parameters for the dark matter component except the usual ones that specify the cosmological context.
MODEL CALIBRATION
We work in the following cosmological context: a concordance flat Λ-CDM cosmology with reduced Hubble constant h=2/3, present
day matter density Ω m=1/3, fluctuation amplitude σ8=0.9, cosmological baryon fraction Ωb=0.04, and primordial power spectrum
index n=1. This is fully consistent with the results provided by the WMAP experiment. Once the cosmology has been fixed, our
model has only two free parameters to play with: the polytropic index, γ, and the input energy by preheating, ∆E. Both parameters
affect only the evolution of gas and not that of the dark-matter component. As shown in Figure 1 below, for a polytropic index
γ=1.2 and a gas preheating energy ∆E=0.55 KeV per particle our model provides a good fit to the observed correlation
between gas luminosity and emission weighted temperature. With those best-fit values for the model parameters, we then
examine in Figure 2 the correlation between halo mass within an overdensity of 500, M500, and gas temperature. From this figure one
can see that the best fitting parameters achieved from Figure 1 also provide the correct normalization and slope of the observed masstemperature correlation.
Figure 1. Luminosity versus emission weighted temperature. The red line shows
the best fit model prediction (see yellow text above), assuming a uniform
metallicity equal to one third solar for the gas. The green line is the prediction
corresponding to a primordial gas composition (null metallicity). Data points and
error bars have been taken from different samples, namely, Markevitch (1998),
Arnaud & Evrard (1999), Helsdon & Ponman (2000), Reiprich & Böhringer
(2002) and Osmond & Ponman (2004). The range of system temperatures covered
by the data goes from poor groups (around 0.5 keV) to rich galaxy clusters
(around 10 keV). As can be seen the dispersion of the data at the lowest
temperatures can not be explained by differences in the metallicity of the
systems, neither by differences in the epoch of formation, the latter being even
smaller than the former and hence not shown in the figure. According to our
model results, the dispersion present in the observations must be mostly attributed
to observational biases and sample incompleteness on one side, and to departures
from spherical symmetry and virialization on the other side.
Figure 2. Mass versus emission weighted temperature. As in Figure 1, the red line
represents our model prediction for the best-fit values reached in the previous
model calibration. The green line is the model prediction for γ=1 and ∆E=0, i.e.,
an isothermal gas with no pre-heating which corresponds to a self-similar gas
evolution. This prediction is compared with the results obtained from
cosmological hydrodynamic simulations (blue line) not including cooling and
heating sources other than gravitation and for which the resulting gas evolution is
self-similar. Points with error bars correspond to observations. Note that although
a self-similar gas yields approximately the same slope as the observations, the
corresponding predicted slope of the correlation is clearly different from that of
the data. In the vertical axis the halo mass, M500, has been measured within a
radius enclosing a mean halo density equal to 500 times the critical density of the
universe.
GAS STRUCTURE IN REAL SPACE
Figure 3. This figure shows, from top to bottom, the temperature,
density, entropy, and total mass fraction profiles of the gas normalized
to their respective central values (except for the gas mass fraction Fx)
and for halos of different masses. The profiles have been computed for
the best-fit model parameters and the cosmology described in the model
calibration above. In the horizontal axis, the radial distance is
represented in units of the halo virial radius, that is, x/c is equal to the
physical distance to halo centre divided by the total halo radius.
INTERGALACTIC HOT GAS
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Hydrostatic equilibrium and spherical symmetry. Gas does not contribute to the halo gravitational potential.
Polytropic equation of state: ρ ∝ T1/(γ-1)
After halo formation, the gas mass grows inside-out and the accreted gas settles at the instantaneous halo virial radius in
hydrostatic equilibrium with the halo gravitational potential.
Virialization of the system after formation leads to an equilibrium state such that the gas specific energy is the same as the
specific energy of the dark matter component. When gas is affected by any source of preheating, we consider that its specific
energy equals the sum of the dark matter specific energy plus the corresponding specific energy increment arising from a
universal energy increment, ∆E.
The present day gas mass fraction of halos at their virial radii is assumed to be equal to the baryon mass fraction of the universe.
Gas cooling and subsequent star formation are neglected.
Free parameters: the polytropic index (see point 2), γ, and the amount of energy injected to the gas by preheating, ∆E.
SURFACE BRIGHTNESS PROFILES
Figure 4. Most observational studies fit the X-ray surface brightness
profile of groups and clusters to the conventional beta-model, SX(R) = S0
(1+[R/rc]2)-3β+1/2, where rc is the X-ray core radius. The fit normally does
not extend beyond the radius R500 enclosing the mass M500 (see Figure 2 for
the definition). Using our model luminosities and the cooling function from
Sutherland & Dopita (1993) we are able to compute the surface brightness
profiles for systems of different X-ray gas temperature. After fitting a beta
model to our results up to a radius of R500 we get the red curves shown in
the left figure which are compared with the observational results obtained
from the samples quoted at the top-left of the figure. Despite the large
dispersion existing in the data, our theoretical curves fare quite
satisfactorily since they appear to roughly track the mean trends that would
be inferred from a by eye fit to the data. Our prediction for the slope
parameter β supports the frequently advocated case for flatter surface
brightness profiles for groups, while for massive clusters/high temperature
systems our model predicts a value close to the canonical 2/3.
GAS ENTROPY AT INNER AND OUTER HALO REGIONS
Figure 5. Gas entropy measured at different distances from halo centre. Comparison of observations (points with error bars) with
model predictions. Red lines correspond to model predictions for γ=1.2 and ∆E=0.55 keV/part, while green lines are the model
predictions for a self-similar gas, γ=1 and ∆E=0. The dotted green lines represent the typical dispersion in the results caused by the
distribution of halo formation times. This illustrates one of the effects of introducing a preheating to the halo gas, which is to make
such dispersion negligible, this is the reason why there are no visible dotted red lines since they are superposed to the solid line.
Right panel. Points with error bars correspond to data from the Finoguenov et al. (2002, ApJ, 578) study of the entropy at a distance
R500 from the halo centre (where c500 is R500 normalized to the characteristic scale radius of the halo density profile). This radius
encloses a halo mass, M500, so that the mean halo density is equal to 500 times the critical density of the universe. As a comparison, the
total halo radius usually encloses a mean density between 100 and 200 times the critical density of the universe. Hence, R500 represents
a region near the outer boundary of the halo. The mass M500 has been used to scale the entropy measurements as well as our model
results as indicated in the vertical axis label (M500,13, represents the mass in units of 1013 solar masses). In principle, when M500 is
nearly proportional to the halo mass, one should expect that in the absence of gas preheating the data points would be distributed along
more or less an horizontal line. To illustrate this behaviour we also plot in green lines the prediction of our model when preheating is
turned off (green line). For massive systems both models give similar results as expected, since preheating has almost no effect in such
systems. Note however that even when no preheating is considered the scaled entropy is not constant. This is because the ratio between
M500 and the halo virial mass is not a constant, but depends on halo concentration. Our model results (red line) reproduce quite well the
mean trend of the data.
Left panel. In this case the data points have been taken from the observational study by Ponman et al. (2003, MNRAS, 343) of the gas
entropy at a radial distance equal to 10% the halo virial radius (0.1c200 in the vertical axis of the plot) as a function of the X-ray
temperature measured a a distance from the system centre equal to 0.3 times the virial radius (0.3c200 in the horizontal axis). Hence,
this measure of the gas entropy involves a region close to the halo centre. According to Ponman et al., the entropy grows from lower to
higher temperature systems, and no entropy floor is apparently observed in the data for low temperature systems. Our model
predictions are consistent with the data for temperatures higher than 1 keV, and predict a logarithmic slope around 1.4 (blue line) but
they also predict the existence of a minimum entropy for systems with temperature around 1 keV. The value of the entropy floor from
our results is very similar to the value inferred in different observational and theoretical studies. In particular, the horizontal longdashed line corresponds to the entropy floor inferred from the observational study of Lloyd-Davies et al. (2000). If the trend predicted
by Ponman et al. for low temperature systems is confirmed by more detailed and extended observational work, it will be certainly in
conflict with our model predictions.
One can see in this figure that for halo masses ≲ 7·1013 solar masses the
normalized profiles are very similar at any radial distance, while below
that mass the similarity is clearly broken, specially at distances to the
halo centre greater than 10% of the halo virial radius. Our best fit model
yields a gas distribution that has constant density, temperature and
entropy cores regardless of halo mass. Note also that our model predicts
that the entropy profiles of massive halos (which are the less affected by
preheating) will have a logarithmic slope of 1.1 (straight dashed line) as
the profiles approach the halo boundary. This is in accordance with
theoretical expectations (e.g., Tozzi & Norman, 2001, ApJ, 546) from
spherical gas mass accretion, and is also consistent with N-body
simulation results. Our model does not predict large isothermal cores,
neither isothermal or isentropic gas profiles even for low-mass systems
representing poor groups. Finally, the total gas mass fraction predicted
by our model for poor groups is smaller than the universal value, while
for massive clusters the gas mass fraction approaches the universal
value as the radial distance to the halo centre increases.
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