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arXiv:astro-ph/0502338v1 17 Feb 2005
Gamma-Rays from Large Scale Structure
Formation and the Warm-Hot Intergalactic
Medium: Cosmic Baryometry with Gamma-Rays
Susumu Inoue
∗†
and Masahiro Nagashima
∗∗‡
∗
Max-Planck-Institut für Kernphysik, Postfach 103980, 69029 Heidelberg, Germany
†
Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str. 1, 85758 Garching, Germany
∗∗
Dept. of Physics, Kyoto University, Sakyo-ku, Kyoto 606-8502, Japan
‡
Dept. of Physics, University of Durham, South Road, Durham DH1 3LE, United Kingdom
Abstract. It is shown that inverse Compton gamma-rays from electrons accelerated in large scale
structure formation shocks can be crucially affected by non-gravitational effects such as radiative
cooling and galaxy formation, with corresponding uncertainties by an order of magnitude in either
the gamma-ray source counts or the extragalactic background contribution. However, this also
implies that such gamma-rays may in the near future provide us with valuable information about
the fraction of cosmic baryons in different forms, particularly the warm-hot intergalactic medium
where the majority of the baryons in the universe are believed to reside. We address this problem in
a simple way through semi-analytic modeling of structure formation shocks which self-consistently
treats merger and accretion shocks.
INTRODUCTION
The majority of the baryons in the universe today are believed to reside in a warm-hot
intergalactic medium (WHIM) at temperatures T ∼ 10
5
− 10
7
K, as a result of shock
heating during the hierarchical buildup of large scale structure in the universe [2, 3].
They are often referred to as ‘missing baryons’, since quantitative measurements of
the WHIM through direct observations are still lacking, hampered by heavy Galactic
obscuration in the relevant extreme UV to soft X-ray bands (notwithstanding important
but fragmentary information from absorption lines that probe only selected lines of
sight [17]). Current indirect estimates of the cosmic fraction of baryons in the WHIM
f
WH
range from ∼20 to ∼70 % [5, 6, 24]. In view of the significance of elucidating
this fundamental component of the universe, dedicated satellite missions such as the
Diffuse Intergalactic Oxygen Surveyor [18] and the Missing Baryon Explorer [4] are
being planned in order to detect emission lines from the WHIM and directly constrain
f
WH
.
On the other hand, the same large scale structure formation (SF) shocks that create
the WHIM may give rise to GeV-TeV gamma-ray emission through nonthermal electron
acceleration and inverse Compton upscattering of the cosmic microwave background.
Such gamma-rays may be observable either as a contribution to the cosmic gamma-
ray background (CGB) [14] or as individual sources [28]. This interesting possibility
has spawned a number of studies using numerical simulations [12, 15] or semi-analytic
methods [9, 10], although most such works had limited their scope to treating purely
gravitational effects. In reality, the global hydrodynamical evolution of intergalactic gas
and the associated gamma-ray emission can be crucially affected by non-gravitational
effects such as radiative cooling (with consequent star formation and feedback) and
photoionization heating.
By considering such non-gravitational effects in a simplified way, we show here
that there should be a nontrivial connection between SF gamma-rays and the baryonic
fraction in the WHIM. The problem is addressed through semi-analytic modeling of SF
shocks based on Monte Carlo merger trees with multiple mergers [22], which allows a
self-consistent treatment of major and minor merger shocks as well as diffuse accretion
shocks. The full details can be found in a forthcoming paper (Inoue and Nagashima, in
prep.).
FORMULATION
An important point in considering nonthermal effects due to SF shocks is that such
shocks can be either strong or weak, with Mach numbers M ranging from very large
ones (≫ 1) for minor mergers between systems with large mass ratios or accretion of
relatively cold gas onto a large object, to values as low as ∼1.5–3 in the case of major
mergers between virialized objects of comparable masses [1, 20, 26]. This implies that
the spectral index of shock accelerated particles p can be either the strong shock limit
of p ≃ 2 or much steeper with p > 2, leading to considerably differenct effects at high
energies [8]. Thus it is imperativeto account for the distribution of shock Mach numbers
appropriately.
One way to address this problem is through full-scale cosmological hydrodynamical
simulations[12, 15, 20]. Here we opt for a semi-analytic approach based on the extended
Press-Schechter (PS) formalism of structure formation [13], which gives a simple yet
reasonably accurate description of the hierarchical gravitational growth of dark matter
halos. In particular, we employ the multiple merger tree algorithm of Somerville &
Kolatt [22], which accurately reproduces the total and conditional halo mass functions
and also accounts for diffuse accretion. At each time step in the merger tree, we also
employ an accurate mass function derived from very high resolution N-body simulation
results [30]. Note that the semi-analytic model of Gabici & Blasi (GB03) [8, 9, 10] is
built on the simpler binary merger tree algorithm of Lacey & Cole [13], which cannot
treat accretion and is known to produce self-inconsistent results when the merger tree is
extended to high redshifts [22]. (Nevertheless, it is foundthat the differences are not very
large at low redshifts, and our results below for SF gamma-rays are in basic agreement
with [9, 10] when appropriate comparisons are made.)
Our basic assumptions are as follows. (1) The adopted cosmological parameters
are Ω
m
=0.3, Ω
Λ
=0.7, Ω
b
=0.044 and h=0.7. The normalization and spectral index of
primordial fluctuations are respectively
σ
8
=0.9 and n=1. (2) At each step, a multi-
ple merger event between more than two halos is pictured as an ensemble of binary
mergers with the primary (i.e. most massive) progenitor. Associated with each binary
pair are two shocks propagating within them. Mass below a certain mass scale (see
below) in the timestep accretes spherically onto the primary. (3) An effective Mach
number is assigned to each merger shock. The temperature of the preshock gas is
the virial value for the relevant progenitor. The relative infall velocity v is given by
v
2
= 2G(M
A+B
)[( f
0
(M
B
/M
A
)+1)/(R
A
+R
B
)−1/2R
AB
], where M
i
and R
i
are the masses
and virial radii for A, B and A+B denoting the two progenitors and merged halo, respec-
tively. This is similar to [8], except that we include a parameter f
0
that is calibrated to
match the simulation results for major mergers [26]. Diffuse accretion shocks are always
strong. (4) The electron injection efficiency is fiducially a fraction
ξ
e
= 0.05 of the en-
ergy dissipated at the shock, i.e. the difference between the post-shock and pre-shock
thermal energies. The injection spectral index p is related to the shock Mach number M
as p = 2(M
2
+ 1)/(M
2
− 1) (test particle assumption). The emitted inverse Compton
spectrum is a broken power-law, with energy indices (p − 1)/2 and p/2 respectively
above and below the cooling break energy where the electron cooling time equals the
shock crossing time, and a high energy cutoff where the cooling time equals the accel-
eration time. Only primary electrons are considered. (5) The gas fraction inside halos
is Ω
b
/Ω
m
when affected solely by gravity. Non-gravitational effects due to radiative
cooling and photoionization heating which can be effective in certain mass ranges are
incorporated in a simplified way as described below.
Two characteristic mass scales are important concerning non-gravitational effects.
One is the scale of the post-reionization Jeans mass (more accurately the filtering mass),
below which gas cannot appreciably cool and collapse inside virialized halos due to
photoionizationheating by the UV background [11]. This may be considered the natural
boundary distinguishing diffuse accretion and clumpy merging and is identified with
the mass resolution scale in our merger tree algorithm. Taking its velocity dispersion
V
acc
as a parameter, a realistic range may be V
acc
≃ 20 − 70 km/s as suggested by
detailed modeling [11]. We assume a fiducial value of V
acc
= 40 km/s, corresponding
to mass 3.0 × 10
10
M
⊙
at z = 0. The other important scale is the maximum cooling
mass above which gas cannot significantly cool because of the reduced cooling function
in the pertinent temperature range; its velocity dispersion is parameterized by V
cut
. In
halos between V
acc
andV
cut
, gas can cool efficiently and condense into stars, i.e. become
galaxies. The observed galaxy luminosity function indicates V
cut
≃ 150− 250 km/s, and
we fiducially takeV
cut
= 200 km/s or 3.7× 10
12
M
⊙
at z = 0 in terms of mass. SF shocks
and associated emission will be suppressed in systems which have converted a large
fraction of its gas into stars, and we treat this effect simply by removing SF shocks in a
fraction f
GF
of halos between V
acc
and V
cut
and account only for mass growth through
their merging.
An interesting connection can be made between our principal parameters V
acc
, V
cut
,
f
GF
, and the present-day fraction of baryons in the universein different forms. Following
[3] in dividing cosmic baryons into four phases, diffuse (T < 10
5
K), condensed (stars
and cold gas), warm-hot (10
5
< T < 10
7
K), and hot (T > 10
7
K), these respectively
relate in our picture to systems with velocity dispersion V < V
acc
, a fraction f
GF
of
V
acc
< V < V
cut
, the rest 1− f
GF
ofV
acc
< V < V
cut
plus a part of V > V
cut
with T < 10
7
K, and the remainder ofV > V
cut
with T > 10
7
K. If we take our fiducial valuesV
acc
= 40
km/s and V
cut
=200 km/s, there is a one to one relation between f
GF
and f
cond
, the
baryon fraction condensed into stars and cold gas. This relation can be quantitatively
evaluated using the semi-analytic galaxy formation model of [16]. In turn, f
cond
can be
related to f
WH
by subtracting the baryon fractions in the diffuse and hot phases. For
example, cosmological simulations including radiative cooling and galaxy formation [3]
indicate a range f
cond
≃ 0.2− 0.4 and f
WH
≃ 0.2− 0.4 at z = 0, which corresponds to
f
GF
≃ 0.6− 0.9. Alternatively, a recent observational census [5] suggests f
cond
≃ 0.1
and f
WH
≃ 0.4, which is consistent with f
GF
≃ 0.4.
A further non-gravitational effect that might be important is feedback (pre-)heating by
supernovae-driven winds or AGN outflows, as indicated by the observed X-ray scaling
relations of groups and clusters ([21] and references therein). Since the details of such
processes are highly uncertain at present, we defer a consideration of these effects to
future work (see [27] for an early, crude discussion).
One important caveat is in order concerning our formulation. In the PS picture, all the
dark matter in the universeis described as being bound insidespherically virializedhalos
of some mass. This is a fairly good approximation, as many comparisons with N-body
simulationsdemonstrate (e.g. [30] and references therein.) However, the same cannot be
said about the gas component. In fact, we have explicitly assumed the fraction of gas
with V < V
acc
to be in diffuse form outside bound halos due to photoionization heating.
It is less clear how much of the WHIM, particularly the gas with V
acc
< V < V
cut
, can be
considered to be inside or outside bound objects. Although hydrodynamical simulations
indicate that a large part of the WHIM arises through shock heating by gravitational
infall onto filamentary or sheet-like structures [2, 3], much of the gas in such structures
may actually be interpreted as residing inside sufficiently small halos if seen at high
enough resolution. Since the essential driving force of WHIM evolution is the gravity of
the dark matter, most of which is indeed in bound form, we have chosen to describe the
WHIM as gas inside bound haloes with the corresponding range of virial temperatures
which do not condense into stars. Just how good such a description (or some alternative,
e.g. [7]) may be can only be judged through future comparisons with detailed numerical
simulations.
RESULTS AND DISCUSSION
Figure 1 shows our results of the SF shock contribution to the CGB for different values
ofV
acc
, V
cut
and f
GF
. To be compared are CGB data from COMPTEL [29] and EGRET,
including both the old Sreekumar et al. (1998; S98) [23] and new Strong, Moskalenko
& Reimer (2004; SMR04) [25] determinations.
We first take V
acc
= V
cut
= 20 km/s, corresponding closest to the situations treated
in numerical simulations, where the sole non-gravitational effect is a temperature floor
of T = 10
4
K [12, 15]. The result accounts for almost all of the S98 CGB, and in fact
exceeds the new SMR04 CGB, indicating either that this case does not represent reality
or that
ξ
e
is significantly less than the fiducial value 0.05. Although this is in more
agreement with the result of [15] than of [12], here we reserve a quantitative judgement,
given the approximate nature of our formulation.
For this and all other cases discussed here, the end result is dominated by minor
merger shocks, with accretion shocks amounting to at most 1% of the S98 CGB. Keep-
ing V
acc
= V
cut
(i.e. no condensation into stars), a larger value decreases the merger
FIGURE 1. Cosmic gamma-ray background for different values of V
acc
, V
cut
and f
GF
as indicated in
the legend, compared with COMPTEL and EGRET (both S98 and SMR04) data.
component and hence the total CGB, while slightly increasing the accretion component;
for smaller values, vice-versa. Taking f
GF
= 1 (complete condensation in the cooling
regime) with V
acc
= 40 km/s fixed, varying V
cut
has a dramatic effect, with the CGB
being suppressed by more than an order of magnitude asV
cut
= 200 km/s is approached.
This can be understood as removing larger and larger galaxy-scale objects, which can
potentially produce strong shocks in minor mergers with cluster-scale objects. Our fidu-
cial set of V
acc
= 40 km/s, V
cut
= 200 km/s leads to ∼ 10 % of the S98 CGB, consistent
with the results of GB03.
Obviously, when condensation occurs only for a fraction f
GF
of objects in the cooling
regime, the reduction is less, and one gets a CGB somewhere between 10 to 100 %
of the S98 CGB. Recalling the above-mentioned connection between f
GF
and f
cond
, the
current uncertainty in f
cond
≃ 0.1− 0.4 allows a range f
GF
= 0.4− 0.9, and the CGB due
to SF shocks cannot be reliably predicted to within an order of magnitude. However, this
points to an interesting possibility of constraining f
cond
and hence f
WH
if the SF shock
contribution to the CGB can be observationally determined. In practice, this requires
removing other contributions (e.g. blazars) to the CGB with good precision, which may
not be an easy task.
A more promising way to constrain f
WH
with gamma-rays may be through the
FIGURE 2. Gamma-ray source counts at 100 MeV and 10 GeV for different values of f
GF
, compared
with the sensitivities of EGRET and GLAST.
statistics of source counts. Figure 2 displays the cumulative source counts due to SF
shocks at energies > 100 MeV and > 10 GeV, for fixed fiducial values of V
acc
and V
cut
and different f
GF
. Again, differences of an order of magnitude can be seen, depending
on how much minor merger shocks are suppressed. For f
GF
= 0.9 (f
cond
≃ 0.25, f
WH
≃
0.25), ∼ 100 and ∼ 10 sources should be observable by GLAST at >100 MeV and
>10 GeV, respectively, while none exists above the EGRET sensitivity at >100 MeV.
For f
GF
= 0.4 (f
cond
≃ 0.1, f
WH
≃ 0.4), the respective numbers are ∼ 600 and ∼ 30 at
>100 MeV and >10 GeV for GLAST, whereas a few are expected for EGRET. The fact
that EGRET actually saw no emission associated with clusters [19] may point to either
f
GF
> 0.4 or
ξ
e
< 0.05. This underlies the need for the electron injection effiency to
be pinned down, preferably through detailed observations of individual sources where
the kinetic energy can be estimated independently. Once this is done, SF gamma-rays
may provide an indirect but very valuable probe of the unknown fraction of baryons
in the WHIM. In fact, a close connection between SF gamma-rays and the WHIM is
not surprising at all, as they both arise from the same large-scale shocks. However, the
quantitative correspondence is a nontrivial one involving the reduction of strong, minor
merger shocks by condensation into stars.
To summarize, we have investigated inverse Compton gamma-rays from large scale
SF shocks including non-gravitational effects with a self-consistent semi-analytic for-
mulation. Radiative cooling and galaxy formation were shown to have crucial impact,
with the predicted CGB contribution and gamma-ray source counts uncertainby an order
of magnitude depending on the fraction of baryons condensing into stars. This in turn
implies that SF gamma-rays may serve as an indirect ‘baryometer’ of the universe and
probe the ‘missing’ fraction of baryons in the WHIM, which is very difficult to measure
directly.
The present work is an example of nonthermal phenomena due to SF shocks where
semi-analytic, PS-based methods can be applied effectively, allowing the exploration of
physical effects in a simple way which is not alwaysthe case withnumerical simulations.
However, in view of the numerous approximations in our formulation, further studies
with simulations are warranted, both to confirm the qualitative trends found here and
to make predictions that are quantitatively more robust. The effects of feedback (pre-
)heating, which have not been treated here, may also be potentially important and need
to be investigated further.
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