A&A 396, 21-30 (2002)
DOI: 10.1051/0004-6361:20021417
M. Bartelmann1 - F. Perrotta2,3 - C. Baccigalupi2,4
1 - Max-Planck-Institut für Astrophysik, PO Box 1317,
85741 Garching, Germany
2 - Lawrence Berkeley National Laboratory, 1 Cyclotron Road,
Berkeley, CA 94720, USA
3 - Osservatorio Astronomico di Padova, Vicolo dell'Osservatorio
5, 35122 Padova, Italy
4 - SISSA/ISAS, via Beirut 4, 34014 Trieste, Italy
Received 1 July 2002 / Accepted 16 September 2002
Abstract
We study the effects of a dark energy component with equation of
state
with constant
on the formation of Cold
Dark Matter (CDM) haloes. We find two main effects: first, haloes
form earlier as w increases, and second, the amplitude of the
dark-matter power spectrum gets reduced in order to remain
compatible with the large scale Cosmic Microwave Background (CMB)
anisotropies. These effects counteract. Using recipes derived from
numerical simulations, we show that haloes are expected to be up to
50% more concentrated in CDM models with quintessence
compared to
CDM models, the maximum increase being reached
for
.
For larger w, the amplitude of the power
spectrum decreases rapidly and makes expected halo concentrations
decrease. Halo detections through weak gravitational lensing are
highly sensitive to halo concentrations. We show that weak-lensing
halo counts with the aperture-mass technique increase by a factor
of
2 as w is increased from -1 to -0.6, offering a new
method for constraining the nature of dark energy.
Key words: cosmology: theory - dense matter - gravitational lensing
In recent years, cosmology has seen increasing observational evidence
for an accelerating phase of the cosmic expansion, most notably
through the observations of distant type Ia supernovae (Perlmutter et al. 1999; Riess et al. 1998). This astonishing evidence motivated
renewed interest in the properties of the energy density ascribed to
the "vacuum''. A vacuum energy component should account for both the
accelerating expansion and for the residual 70% of the energy
density required for reconciling the geometrical flatness required by
Cosmic Microwave Background (CMB) observations (De Bernardis et al. 2002; Lee et al. 2001; Halverson et al. 2002) with the evidence of
a low-density universe with
(Percival et al. 2001).
While one of the historical candidates for such an energy density is the cosmological constant, introduced as a simple geometrical term in Einstein's equations, it is well known that it leads to serious and unsolved theoretical problems. The exceedingly low value of the vacuum energy density today, compared to that allowed by the most plausible theories of the early stage of the Universe, motivated the introduction of a more general concept now widely known as "dark energy''.
Preceding the evidence for cosmic acceleration, a generalisation of the cosmological constant by means of a scalar field, now known as "quintessence'', was proposed (Wetterich 1988; Ratra & Peebles 1988). In this class of models, the dark energy is supposed to reside mostly in the potential energy of the field, which interacts only gravitationally with ordinary matter. The evolution is described by the ordinary Klein-Gordon equation. If the potential is flat enough, or if the motion of the field along its trajectory is sufficiently slow, a cosmological constant-like behaviour can be mimicked by the scaling of the energy density of the quintessence field.
For general forms of the scalar field potential, there exist attractor trajectories for the evolution of the background expectation value of the field. These trajectories are known as "tracking'' (Steinhardt et al. 1999) and "scaling'' (Liddle & Scherrer 1999) solutions. They have been shown to alleviate, at least at a classical level, the fine-tuning required in the early Universe, when the typical energy scales were presumably comparable to the Planck scale, to generate a vanishing relic vacuum energy as it is observed today, 120 orders of magnitude smaller. However, these scenarios are not able to solve the coincidence problem, i.e. why we are living in the epoch in which dark energy and matter have roughly the same energy density. Despite some attempts at addressing this issue (Tocchini-Valentini & Amendola 2002; Chiba 2001; Armendariz-Picon et al. 2001; Dodelson et al. 2000), it remains one of the greatest puzzles of modern cosmology.
For constraining the nature of the dark energy, an important step would be accomplished if parameters could be constrained which capture its most essential features. In particular, if the dark energy is modelled as a quintessence field in the tracking regime, the simplest description of its properties will require the use of only two parameters, i.e. its present energy density and the ratio between pressure and energy density in its equation of state. Generally, this ratio depends on time, as implied by the evolution according to the equation of motion. However, it can be shown that, in most simple models of quintessence involving an inverse power-law potential, the effect of a time variation of the equation of state can be neglected at low redshifts, when the field has settled on its tracking trajectory. In this case, the equation of state is simply related to the power with which the potential depends on the field itself. This simplification allows constructing a scheme for describing the dark energy behaviour at redshifts where interesting cosmological effects arise, such as the effect on the magnitude-redshift relation of type Ia supernovae (Perlmutter 1999; Riess 1998) and the effect on gravitational lensing of distant galaxies and quasars (Futamase & Yoshida 2001).
Even though the dark energy dynamics has a geometrical effect on acoustic features of the CMB anisotropy (Baccigalupi et al. 2002; Doran et al. 2002; Corasaniti & Copeland 2002), it is now commonly accepted that the most interesting properties of a dark energy component have to be probed by looking at processes occurring at low redshifts when it starts dominating the cosmic expansion.
Thus, one of the most powerful probes of the quintessence field
results from its effects on cosmic structure formation, most notably
on the background cosmology and the evolution of individual collapsing
overdensities. First, a dark energy component affects the matter
density of the background in which haloes form. Second, the amplitude
of matter perturbations is sensitive to the presence of a dynamical
vacuum energy, through the normalisation of the matter power spectrum
to the large, unprocessed, scales probed by the large-scale CMB
anisotropies. Third, changes in the background matter density induced
by a dark energy component can seriously affect characteristic
properties of collapsing structures. As shown by okas & Hoffmann
(2002), a substantial quintessence component changes the
characteristic density of a forming dark matter halo.
In this paper, we study how quintessence affects the concentration of dark-matter haloes, and resulting changes in their weak-lensing efficiency. Weak gravitational lensing provides a powerful tool for mapping the large-scale mass distribution (see Mellier 1999a and Bartelmann & Schneider 2001 for reviews), and the potential impact of dark energy on the weak lensing convergence power spectrum has already been recognised (see Mellier 1999b; Huterer 2002). We show in this paper that an energy density component with negative pressure affects weak lensing by dark-matter haloes not only through changes in the global properties of the Universe, but also by modifying their internal density concentration. The main idea is that dark energy affects structure growth and thus the time of halo formation. Since halo concentrations reflect the density of the Universe at their formation epoch, this affects halo mass distributions, and weak lensing provides methods for quantifying respective changes.
The paper is organised as follows. In Sect. 2, we describe the main effects of the dark energy on the cosmological growth factor, volume elements, and the normalisation of the dark-matter power spectrum. In Sect. 3 we compute the impact on halo concentration. In Sect. 4 we predict resulting effects on the weak-lensing aperture mass statistics, and Sect. 5 contains our conclusions.
We model the dark energy as a spatially homogeneous component,
labelled Q, with constant equation of state parameterised by the
ratio between the pressure
and the energy density
,
.
We neglect a
possible time variation of w, as well as any effects possibly due to
spatial inhomogeneities of the dark energy. Indeed, at least in most
models proposed so far, the relevant cosmological effects of the dark
energy compared to a cosmological constant are mainly related to its
effective equation of state at redshifts when it is relevant for
cosmic expansion, say
,
as we already noted in the
introduction. Neglecting inhomogeneities of the dark energy is
justified in the present context since they are likely to show
relativistic behaviour on sub-horizon cosmological scales; indeed, the
effective mass of the vacuum component, of the order of the critical
density today, is extremely light compared to any other known massive
particle, so that quintessence clustering occurs only on scales larger
than or equal to the horizon size (Ma et al. 1999). It can also be
shown formally that a minimally coupled quintessence field has a
relativistic effective sound speed (Hu 1998), so that its fluctuations
are damped out on the scales in which we are interested here.
Assuming the equation of state
,
the
adiabatic equation implies
The quintessence term in (2) has two immediate consequences relevant for our purposes. First, the way how density inhomogeneities grow is modified, and second, the cosmic volume per unit redshift changes.
In linear theory, the density contrast
of matter
perturbations grows according to
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Figure 1:
Growth factor D+(z) as a function of redshift for five
different cosmological models as indicated. The growth factor is
normalised to unity at the present epoch, and divided by the scale
factor to emphasise the differences between the models. With
increasing w, the growth factor increases towards its value for
the open model with
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The solid curve in Fig. 1 shows the growth factor for a
model universe with a cosmological constant (w=-1). Going back in
time, D+/a rises from unity to 1.3 and turns flat near
.
This means that structures start forming
earlier in this model compared to an Einstein-de Sitter model, but the
growth slows down considerably at redshifts smaller than
.
For
the low-density open model without quintessence (labelled
), D+/a keeps rising as z increases. At
redshift 5, for instance, the amplitude of structures is 1.4 times
higher than in the cosmological-constant model. Increasing winterpolates between the open and the flat model with cosmological
constant. Thus, keeping
and
fixed,
structures form earlier for larger values of w, approaching the
growth behaviour for low-density open models without quintessence or
cosmological constant.
A similar interpolation is seen in the behaviour of the cosmic volume
per unit redshift. Figure 2 shows
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Figure 2:
Cosmic volume in units of the Hubble volume between redshift
zero and z for six different cosmological models as
indicated. With increasing w, the cosmic volume decreases towards
its value for the open model with
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Here and below, we keep the cosmological parameters fixed at
,
.
The Hubble constant is
with h=0.7.
In order to describe the formation of dark-matter haloes, we need to
specify the power spectrum of dark-matter fluctuations. We choose the
cold dark matter (CDM) model, whose transfer function was given by
Bardeen et al. (1986), and we set the index of the primordial power
spectrum to the Harrison-Zel'dovich value of n=1. Then, the power
spectrum has two free parameters, the shape parameter
which
locates its maximum, and the amplitude. For the shape parameter, we
assume
as suggested by theory and in agreement with
observations of the galaxy power spectrum. For our model for dark
energy, the transfer function by Bardeen et al. is applicable because
dark energy does not cluster on the relevant scales, and the shape
parameter
is set by the scale of matter-radiation equality,
which is unaffected by w (cf. Wang & Steinhardt 1998).
The amplitude of the power spectrum needs to be chosen so that certain observations can be reproduced. It is an important constraint that the abundance of rich galaxy clusters in our cosmic neighbourhood be reproduced. Since the galaxy-cluster mass function falls very steeply at the high-mass end, small changes in the amplitude of the power spectrum lead to large changes in the cluster number density, thus the amplitude is in principle well constrained by the cluster abundance. However, for that normalisation procedure, rich galaxy clusters are identified by their X-ray emission, thus the reliability of the normalisation depends on how well models can describe the X-ray properties of the intracluster gas.
We saw before that structure starts forming earlier in quintessence
models with w>-1 compared to models with cosmological
constant. Galaxy clusters forming earlier are hotter because of the
higher mean density of their surroundings. Reproducing the current
number density of X-ray selected clusters thus requires a
power-spectrum amplitude which decreases with increasing w (Wang &
Steinhardt 1998). Figure 3 shows their results, expressing
the power-spectrum amplitude in terms of the rms fluctuation
amplitude
on a physical scale of
.
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Figure 3:
The normalisation of the power spectrum, expressed in terms
of ![]() ![]() ![]() ![]() ![]() |
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The power spectrum of density perturbations can be normalised by the CMB anisotropies on large angular scales, corresponding to physical scales which were larger than the horizon at decoupling. The only available data on those scales are those of the Differential Microwave Radiometer (DMR) experiment on board the COsmic Background Explorer satellite (COBE, Bennett et al. 1996), but much improved data are being taken by the Microwave Anisotropy Probe (MAP, http://map.gsfc.nasa.gov/), and will be taken by the Planck satellite (http://astro.estec.esa.nl/SA-general/Projects/Planck/). Sub- degree CMB anisotropies have been measured by several experiments (De Bernardis et al. 2001; Lee et al. 2001; Halverson et al. 2002). However, on these angular scales, the main effect of increasing w above -1 is to rigidly shift acoustic peaks toward larger angular scales (see Baccigalupi et al. 2002 and references therein), having almost no effect on the overall normalisation of the spectrum.
Normalising the power spectrum according to the COBE measurements
fixes the amplitude of the power spectrum on its large-scale end. As
w increases at fixed
and
,
dark energy
dominates cosmic expansion earlier compared to a model with
cosmological constant having the same energy density today. This has
the effect of enhancing the dynamics of the gravitational potential
due to the change in the cosmic equation of state, thus increasing the
Integrated Sachs-Wolfe (ISW) effect on COBE scales. While this trend
is gentle for
,
it steepens for larger w, as shown by
the dotted line in Fig. 3.
This non-linear behaviour of
reflects the sensitivity of
the ISW effect (and thus the normalisation) to the redshift
at which dark energy starts dominating over matter. It
is easy to show that this is determined by the power law
.
We recall that,
although we consider values of w up to -0.4 for the sake of
generality, values much higher than -0.6 are not interesting in the
framework of dark-energy models. Indeed, w larger than -0.48 is
insufficient to provide cosmic acceleration if
as we assume in this work. In addition, the sharp decrease in
Fig. 3 due to the ISW leads to a decrease of the acoustic
peaks in the CMB power spectrum below the level observed by
experiments operating on sub-degree angular scales.
In order to avoid the uncertainties in
due to the
uncertainties in modelling the X-ray cluster population, we choose the
COBE normalisation for our study. We adopt the normalisation method by
Bunn & White (1997), which has a
accuracy. This procedure
exploits a maximum likelihood approach for reproducing the measured
CMB anisotropy power once the sky regions affected by the Galactic
signal have been cut out. The CMB anisotropy can be expressed as a
line-of-sight integral of the perturbation power spectrum weighted
with suitable geometric functions, implemented in the CMBFAST code
(Seljak & Zaldarriaga 1996). For taking the dark energy component
into account, we use a modified version of CMBFAST (see Baccigalupi et al. 2000). Since halo properties are determined by
the small-scale end of power spectrum while the COBE normalisation
fixes its large-scale end, quantitative results will sensitively
depend on the index n of the power spectrum.
An entirely different and perhaps more direct way of normalising the power spectrum has become feasible recently. Large-scale density fluctuations differentially deflect light on its way from distant sources to us. The gravitational tidal field of the matter inhomogeneities coherently distorts the images of faint background galaxies. Albeit weak, this cosmic shear effect has recently been measured successfully by several groups, whose results agree impressively although different telescopes, observational parameters and data-reduction techniques were used (van Waerbeke et al. 2000; Bacon et al. 2000; Kaiser et al. 2000; Wittman et al. 2000; Maoli et al. 2001; van Waerbeke et al. 2001). Since gravitational lensing depends only on the matter distribution and not on its composition or physical state, cosmic shear should provide one of the cleanest ways for constraining the power-spectrum amplitude.
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Figure 4:
The squared cosmic shear
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Cosmic-shear measurements have been shown to agree very well with the
expectations in a CDM universe. Fixing the shape parameter to
,
the measurements require
and
at 95% confidence (van Waerbeke et al. 2001). Note,
however, that these constraints depend on the nonlinear evolution of
the power spectrum which has some uncertainties.
This result is almost completely insensitive to w in quintessence
models. In Fig. 4, we show the rms cosmic shear in
apertures of radius
as a function of
for four models
differing by w, as indicated. The power spectrum was normalised to
reproduce the COBE measurements, and its nonlinear evolution was
approximated using the fitting formulae by Peacock & Dodds (1996).
Following van Waerbeke et al. (2001), we adopted the source redshift
distribution shown in (9) below with parameters z0=0.8and
.
The curves are fully compatible with the current
cosmic-shear measurements. Moreover, they are much closer to each
other than the typical uncertainty of the measurements, which is
mainly due to the fact that cosmic shear is most sensitive to
structures below redshift
0.5 where the differences between the
models are small. This allows two conclusions. First, the COBE
normalisation is not in conflict with the measured cosmic shear for
all interesting values of w, otherwise the curves would disagree
with the measurements; and second, the constraint on
derived from current cosmic-shear data is independent of wbecause the curves do not significantly change with w.
Numerical simulations of cosmic structure formation show consistently
that the density profiles of dark-matter haloes can be described by a
two-parameter family of models. Far outside a scale radius
,
the profiles fall off proportional to r-3, while
they are cuspy but considerably flatter well within
.
The exact inner profile slope is under debate. The
second free parameter besides the scale radius is a characteristic
density scale
.
We adopt the density profile
suggested by Navarro et al. (1995),
Numerically simulated haloes turn out to be the more concentrated the less massive they are. Since less massive haloes form earlier than more massive ones in hierarchical models of structure formation, this is interpreted assuming that the central density of a halo reflects the mean cosmic density at the time when the halo formed (e.g. Navarro et al. 1997).
Several algorithms based on this assumption have been suggested for
describing the concentration of dark-matter haloes. Originally,
Navarro et al. (1997) devised the following approach. A halo of mass
M is first assigned a collapse redshift
defined as
the redshift at which half of the final halo mass is contained in
progenitors more massive than a fraction
of the final
mass. Then, the density scale of the halo is assumed to be some factor
C times the mean cosmic density at the collapse redshift. They
recommended setting
and
because
their numerically determined halo concentrations were well fit
assuming these values.
Bullock et al. (2001) noticed that halo concentrations change more
rapidly with halo redshift than the approach by Navarro et al. (1997)
predicts. They suggested a somewhat simpler algorithm. Haloes are
assigned a collapse redshift defined such that the non-linear mass
scale at that redshift is a fraction
of the final halo
mass. The halo concentration is then assumed to be a factor K times
the ratio of the scale factors at the redshift when the halo is
identified and at the collapse redshift. Comparing with numerical
simulations, they found
and K=4.
Yet another algorithm was suggested by Eke et al. (2001). They
assigned the collapse redshift to a halo of mass M by requiring that
the suitably defined amplitude of the linearly evolving power spectrum
at the mass scale M equals a constant
.
Numerical results are well represented setting
.
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Figure 5:
Halo concentration parameters c as functions of halo mass
M200. Results obtained from three different prescriptions for
calculating the concentration are shown; these are the prescriptions
from Navarro et al. (NFW), Bullock et al. (B) and Eke,
Navarro & Steinmetz (ENS). Curves are shown for the ![]() ![]() |
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Since we consistently use R200 and M200 for parameterising
haloes, we need to convert masses and concentration parameters from
the slightly different definitions introduced by Bullock et al. and
Eke et al. In particular, this requires us to iteratively compute the
concentration parameter according to our definition. Moreover, the
algorithms implicitly use the mean overdensity of virialised haloes,
,
and the linear overdensity of collapsed haloes,
.
These parameters depend on cosmology and on the
dark energy parameter w. We compute them using the formulae given in
okas & Hoffmann (2002).
Although halo concentrations produced by these different algorithms
differ in detail, they have in common that haloes forming earlier are
more concentrated. Taken together with the earlier result that haloes
form earlier in dark energy models with w>-1 than in CDM
models, this implies that haloes are expected to be more concentrated
in models with w>-1. Figure 5 illustrates this. Halo
concentrations computed with the three different algorithms are
plotted as functions of halo mass for redshift zero. Since less
massive haloes start forming earlier than more massive ones in
hierarchical models like CDM, the concentration decreases with halo
mass. Curves are shown for models with w=-1 and w=-1/2, keeping
all other parameters fixed. At
,
for instance, halo concentrations are approximately 50% higher in the
dark energy compared to the
CDM model.
Concentrations of haloes with fixed mass M identified at z>0 tend to be smaller than at redshift zero. This does not contradict the finding that haloes forming earlier are more concentrated because haloes of a given mass M at z>0 have larger mass at redshift zero, thus the change of halo concentrations with redshift for fixed mass reflects their change with mass for fixed redshift. Likewise, the trend of increasing concentrations with increasing w remains also at higher redshifts.
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Figure 6:
Halo concentrations obtained from the algorithm described by
Bullock et al. as functions of halo mass M200 for five
different cosmological models as indicated. With increasing w, the
concentration increases until
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Figure 6 shows halo concentrations as functions of mass
derived from the algorithm suggested by Bullock et al. for five
different choices of w, keeping all other parameters fixed. As wincreases away from -1, concentrations first increase for all halo
masses shown. A maximum is reached for .
As w is raised
further, concentrations decrease rapidly. This is an effect of the
power-spectrum normalisation. As w increases,
must
decrease because otherwise the normalisation constraints would be
violated; either there would be too many hot galaxy clusters, or the
secondary CMB anisotropies caused by the integrated Sachs-Wolfe effect
would exceed the COBE measurements. As
is lowered, haloes
form later, thus counter-acting the earlier increase of the growth
factor in quintessence models. Adopting the COBE normalisation, the
maximum effect is achieved just before the integrated Sachs-Wolfe
effect sets in strongly at
,
cf. Fig. 3.
The impact of dark energy on halo concentrations should cause an impact on many observable quantities. As an illustration, we will now describe how the number of haloes detectable through their weak gravitational lensing effect change with w.
The gravitational tidal field of individual sufficiently massive haloes imprints coherent distortions on the images of faint background galaxies in their neighbourhood. Haloes can thus be detected searching for their characteristic signature on the appearance of the background galaxy population. A sensitive and convenient technique for halo detection, the aperture mass technique, has been suggested by Schneider (1996; see also Kruse & Schneider 1999).
Consider a circular aperture of angular radius .
The aperture
mass is defined as a weighted integral of the lensing convergence
across the aperture,
Through
,
the aperture mass (7) depends
on the source redshift. We compute mean aperture masses by averaging
over the normalised source-redshift distribution
If the weight function
is compensated,
Halo detection can then proceed as follows. An aperture of given
radius is shifted across a wide and sufficiently deep field. At all
aperture positions, the aperture mass is determined. Potential haloes
are located where
exceeds a certain threshold.
The signal-to-noise ratio of an aperture-mass measurement is given by
with the dispersion
Convolving the projected NFW density profile (cf. Bartelmann 1996)
with the weight function
,
the aperture mass of
NFW haloes as a function of halo mass,
is
easily computed. Assuming further a halo mass function
,
we can calculate the number of haloes per unit
mass and redshift whose aperture mass is sufficiently high for the
signal-to-noise ratio
to exceed a given threshold
.
We choose the mass function suggested by
Sheth & Tormen (1999), which is a variant of the Press-Schechter
(1974) mass function which well reproduces the mass function found in
numerical simulations. We take into account that our definition of
mass differs slightly from Sheth & Tormen's in that we use the mass
enclosed by a sphere in which the mean density is 200 times the
critical rather than the mean density.
In calculating the number density of haloes of mass M at redshift
z which produce a significant weak-lensing signal,
,
we have to take into account that a
signal-to-noise threshold
for the
weak-lensing signal does not correspond to an equally sharp threshold
in halo mass because of the scatter in the aperture mass which is
caused by the shot noise from the discrete background galaxy positions
and their intrinsic ellipticity distribution. A halo of mass M has a
certain probability
to produce an aperture mass
,
which we model as Gaussian,
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Figure 7:
Contours in the mass-redshift plane showing the number
density of such haloes which are capable of producing a significant
weak-lensing signal with the aperture-mass technique. The contours
are drawn at a single, arbitrarily chosen level corresponding to a
halo density of
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The figure shows two sets of contours, inner and outer, whose levels
are 10-8.5 and
,
respectively. Each set
has four contours for different values of w, as indicated in the
figure. All other cosmological parameters are kept fixed,
i.e.
and
.
The solid contour is
for the
CDM model, for which w=-1. The dotted and
short-dashed contours show that the region in the mass-redshift plane
occupied by significantly lensing haloes widens as w increases from
-1 to -0.6. If w is increased further, this region shrinks
considerably, as the long-dashed contour shows. This illustrates the
competition between the two effects outline above: haloes grow earlier
and are thus more concentrated in dark-energy models with w>-1, but
the integrated Sachs-Wolfe effect reduces the power-spectrum
normalisation required by the COBE-DMR data. The maximum extent of the
contours in the mass-redshift plane is reached just before the strong
decrease in
near
illustrated in
Fig. 3.
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Figure 8:
The redshift distribution of weak-lensing haloes is shown for
four different cosmological models, as indicated. As w changes,
the peak remains near ![]() ![]() |
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Figure 8 shows the redshift distribution of weak-lensing
haloes for the same four dark energy models used for
Fig. 7. The curves in Fig. 8 are thus integrals
over mass of the distributions in Fig. 7,
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Figure 9:
Total number of significant weak-lensing haloes per square
degree as a function of w. The curve reaches a peak near
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Finally, we show in Fig. 9 the total expected number of
weak-lensing haloes per square degree as a function of w. The peak
is reached with 17 haloes per square degree at w=-0.6, which
is almost a factor of two higher than for
CDM models. Note
also that the increase is roughly linear with w up to the maximum.
For larger values of w, the halo number drops steeply. However, as
already stressed, these cases are not interesting for cosmology since
they do not produce cosmic acceleration as required by observations.
We investigated the expected properties of dark-matter haloes in dark
energy cosmologies. For our purposes, the essential features of such
models are captured describing the dark energy as a density component
with negative pressure
,
where
is a constant. The dark energy density
is determined
by its present value in units of the critical density,
.
In agreement with results from observations of
the CMB, we focus on models which are spatially flat,
,
have a matter density parameter
,
and a Hubble constant of h=0.7.
The modified background dynamics in dark-energy models has two
immediate consequences. First, the growth factor is changed, which
determines how structures grow against the expanding background. In
models with fixed parameters
and
,
structures form earlier if w is larger. Second, under equal
circumstances, the cosmic volume shrinks as w increases. Dark
energy models thus "interpolate'' between low-density, spatially flat
models with cosmological constant and low-density, open models.
For our purposes, we can neglect the clustering of the quintessence
field and assume that the dark-matter power spectrum is given by the
common CDM spectrum. We
take the shape parameter to be given by
and
normalise the spectrum such that the COBE-DMR measurements of CMB
fluctuations on large angular scales are reproduced. This implies a
third cosmological consequence. As w increases, the gravitational
potential of matter fluctuations evolves more rapidly along a given
line of sight. Secondary anisotropies in the CMB caused by the
integrated Sachs-Wolfe effect thus grow in amplitude. Keeping the
total fluctuation amplitude fixed to the COBE-DMR data thus requires
the amplitude of the primordial fluctuations to decrease. Expressing
the power-spectrum normalisation by
,
this implies that
must decrease as w increases. The decrease is gentle for
and steepens as w increases further.
These findings have two counter-acting effects on the evolution of
dark-matter haloes. First, haloes forming earlier are more
concentrated because their core density reflects the density of the
background universe at their formation time. Since structures form
earlier in dark energy models as w is increased, haloes are expected
to become more concentrated as w grows. Second, the decrease of
with increasing w has the opposite effect on the halo
formation time and indirectly on halo concentration. However,
cosmologically interesting dark-energy equations of state must yield
cosmic acceleration today and require
.
Within that
range, the first effect is dominant, and the overall behaviour is
monotonic.
Ideally, extensive, high-resolution numerical simulations would be necessary for quantifying the net result of these two effects. However, simple algorithms for calculating halo concentrations have already been derived from existing numerical simulations. We used them for our work, assuming that they are also valid with the modification of Friedmann's equation caused by the introduction of dark energy instead of a cosmological constant.
We used three different recipes for computing halo
concentrations. Albeit differing in detail, they agree in
concept. Haloes are assigned a formation epoch, essentially requiring
that a certain fraction of the final halo mass has already collapsed
into sufficiently massive progenitors. The characteristic density of
the haloes is then taken to be proportional to the mean background
density of the universe at the halo formation epoch. We showed that
all three recipes lead to the result that haloes are expected to be
increasingly more concentrated as w grows in quintessence models,
showing that the effect of their earlier growth is stronger than the
effect of decreasing .
This holds for
and
reverses for larger w because the integrated Sachs-Wolfe effect then
requires a steep decrease in
.
The particular recipe for
computing halo concentrations described by Bullock et al. implies that
haloes should be
50% more concentrated for w=-0.6 than for
w=-1, where the increase is roughly linear with w.
Finally, we described that halo searches using weak-lensing techniques
are sensitive to halo concentrations. Using the Sheth-Tormen
modification of the Press-Schechter mass function for quantifying the
halo population in mass and redshift, and the aperture mass technique
for quantifying the weak-lensing effects of haloes, we showed that the
expected number density on the sky of haloes causing 5-weak-lensing detections approximately doubles as w increases from
-1 to -0.6, where the increase is linear with w. Our results
indicate that halo concentrations may be a sensitive probe for the
dark-energy equation of state, and that gravitational lensing may
provide the observational tools for applying that probe.
Note, however, that we did not allow variations in some cosmological
parameters which may also change the number of weak-lensing haloes. In
particular, an effect may arise from varying the index n of the
dark-matter power spectrum because it directly affects the
determination of .
On the other hand, our approach here is
to characterise the main effects of dark energy on halo formation and
to propose weak lensing studies as a tool for constraining the dark
energy itself, assuming the main cosmological parameters will be
measured by independent observations like those of the CMB.
Thus, weak lensing turns out to be a powerful tool not only for mapping the distribution of matter in the Universe, but also for probing fundamental dark-energy properties. Currently, weak lensing observations do not allow detailed reconstructions of halo density profiles (Mellier 2001; Clowe et al. 2000; Mellier & van Waerbeke 2001), mainly because of the resolution limit due to the finite number density of background galaxies.
On the other hand, interesting new perspectives have been opened by several recent wide-field cosmic-shear studies (Bacon et al. 2000; Bacon et al. 2002; Wittman et al. 2000; van Waerbeke et al. 2000; Kaiser et al. 2000). Future weak lensing surveys will cover even larger fields and, thanks to the improved control of systematic errors, they will allow tighter constraints on the cosmological parameters. Besides the wide field telescopes which are currently in their project study phases, we specifically mention the "dark matter telescope'' LSST (http://dmtelescop.org, proposed to scan a 7 square-degree sky field), the VISTA survey (http://www.vista.ac.uk), and the SNAP satellite (http://snap.lbl.gov), whose weak lensing survey will cover an area of 300 square degrees, resulting in a very wide field survey with excellent image quality and depth.
Acknowledgements
We thank Uros Seljak and Simon White for valuable discussions.