Mon. Not. R. Astron. Soc. 000, 000–000 (0000) Printed 14 January 2014 (MN LATEX style file v2.2)
Probing decisive answers to dark energy questions from
cosmic complementarity and lensing tomography
Mustapha Ishak
Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544, USA
arXiv:astro-ph/0501594v2 15 Jul 2005
14 January 2014
ABSTRACT
We study future constraints on dark energy parameters determined from several com-
binations of cosmic microwave background experiments, supernova data, and cosmic
shear surveys with and without tomography. In this analysis, we look in particular for
combinations of experiments that will bring the uncertainties to a level of precision
tight enough (a few percent) to answer decisively some of the dark energy questions.
In view of the parameterization dependence problems, we probe the dark energy using
two variants of its equation of state w(z), and its energy density ρde (z). For the latter,
we model ρde (z) as a continuous function interpolated using dimensionless parameters
Ei (zi ) ≡ ρde (zi )/ρde (0). We consider a large set of 13 cosmological and systematic
parameters, and assume reasonable priors on the lensing and supernova systematics.
For CMB, we consider future constraints from 8 years of data from WMAP, one year
of data from Planck, and one year of data from the Atacama Cosmology Telescope
(ACT). We use two sets of 2000 supernovae with zmax = 0.8 and 1.5 respectively,
and consider various cosmic shear reference surveys: a wide ground-based like survey,
covering 70% of the sky, and with successively 2 and 5 tomographic bins; a deep space-
based like survey with 10 tomographic bins and various sky coverages. The one sigma
constraints found are {σ(w0 ) = 0.086, σ(w1 ) = 0.069}, {σ(w0 ) = 0.088, σ(wa ) = 0.11},
and {σ(E1 ) = 0.029, σ(E2 ) = 0.065} from Planck, supernovae and the ground-based
like lensing survey with 2 bins. When 5 bins are used within the same combination the
constraints reduce to {σ(w0 ) = 0.04, σ(w1 ) = 0.034}, {σ(w0 ) = 0.041, σ(wa ) = 0.056},
and {σ(E1 ) = 0.012, σ(E2 ) = 0.049}. Finally, when the deep lensing survey with
10% coverage of the sky and 10 tomographic bins is used along with Planck and the
deep supernovae survey, the constraints reduce to {σ(w0 ) = 0.032, σ(w1 ) = 0.027},
{σ(w0 ) = 0.033, σ(wa ) = 0.04}, and {σ(E1 ) = 0.01, σ(E2 ) = 0.04}. Other coverages of
the sky and other combinations of experiments are explored as well. Although some
worries remain about other systematics, our study shows that after the combination
of the three probes, lensing tomography with many redshift bins and large coverages
of the sky has the potential to add key improvements to the dark energy parameter
constraints. However, the requirement for very ambitious and sophisticated surveys in
order to achieve some of these constraints or to improve them suggests the need for
new tests to probe the nature of dark energy in addition to constraining its equation
of state.
Key words: cosmology: theory – dark energy – gravitational lensing – large-scale
structure of universe
1 INTRODUCTION
One of the most important and challenging questions in cosmology and particle physics is to understand the nature of the
dark energy that is driving the observed cosmic acceleration, see e.g. (Weinberg (1989); Carroll et al. (1992); Turner (2000);
Sahni & Starobinsky (2000); Padmanabhan (2003); Ishak (2005)). An important approach to this problem is to constrain
c 0000 RAS
, 2 M. Ishak
the properties of dark energy using cosmological probes. This would provide measurements that would allow one to test
various competing models of dark energy. However, due to a high degeneracy within a narrow range of the parameter space,
constraining conclusively dynamical dark energy models is going to be a difficult goal to achieve, and much effort and strategy
will be needed. Ultimately, a combination of powerful cosmological probes and tests will be necessary.
In this paper, we study how dark energy parameters are constrained from different combinations of cosmic microwave
background (CMB) experiments, supernovae of type Ia (SNe Ia) data, and weak lensing surveys (WL) with and without
tomography. In particular, we look for combinations of experiments that will be able to constrain these parameters well
enough to settle decisively some of the dark energy questions, say to a few percent. When CMB measurements constrained
the total energy density to ΩT = 1.02 ± 0.02 to a one sigma level (Spergel et al. (2003); Bennet et al. (2003)), it became
generally more accepted that spatial curvature is negligible. Thus, an uncertainty of a few percent on dark energy parameters
could be set as a reasonable goal. Of course, one should bear in mind that it will remain always possible to construct dynamical
dark energy models that are indistinguishable from a cosmological constant within these limits, and therefore, one needs to
resort to cosmological tests beyond the equation of state measurements. A better scenario providing a decisive answer would
be one in which one could show that dark energy is clearly not a cosmological constant.
It is certainly wise to probe the nature of dark energy using gradual steps. However, in both the scenarios mentioned
above, one should keep in mind that the results and conclusions obtained from an analysis where the equation of state is
assumed constant are subject to changes if the equation of state is allowed to vary with the redshift. In this paper, we consider
dark energy with a varying equation of state.
We chose the combination CMB+SN Ia+WL as various studies have already shown that supernovae type Ia constitute a
powerful probe of dark energy via distance-redshift measurements, see for example (Riess et al (1998); Garnavich et al. (1998);
Filippenko & Riess (1998); Perlmutter, et al. (1999); Perlmutter S., et al. (1997); Riess et al (2000); Riess et al (2001))
(Tonry et al. (2003); Knop et al. (2003) ; Barris et al,, (2004); Riess et al. (2004)). Also, several parameter forecast studies
have shown that combining constraints from weak gravitational lensing with constraints from the CMB is a powerful combina-
tion to constrain dark energy; see, e.g. (Hu (2001); Huterer (2002); Huterer & Turner (2001); Benabed & Van Waerbeke (2003))
(Abazajian & Dodelson (2003); Refregier et al. (2003); Heavens (2003) ; Simon et al (2003); Jain & Taylor (2003))
(Bernstein & Jain (2004); Song & Knox (2004)). Importantly, weak lensing measurements are sensitive to both the effect of
dark energy on the expansion history and its effect on the growth factor of large-scale structure. Furthermore, in addition
to tightening the constraints, using independent probes will allow one to test the systematic errors of each probe, which are
serious limiting factors in these studies. For each of these probes, much data will be available in the near and far future.
For WL, there are many ongoing, planned and proposed surveys, such as the Deep Lens Survey (http://dls.bell-labs.com/)
(Wittman et al.(2002)); the NOAO Deep Survey (http://www.noao.edu/noao/noaodeep/); the Canada-France-Hawaii Tele-
scope (CFHT) Legacy Survey (http://www.cfht.hawaii.edu/Science/CFHLS/) (Mellier et al.(2001)); the Panoramic Sur-
vey Telescope and Rapid Response System (http://pan-starrs.ifa.hawaii.edu/); the Supernova Acceleration Probe (SNAP;
http://snap.lbl.gov/) (Rhodes et al.(2003); Massey et al.(2003); Refregier et al.(2003)); and the Large Synoptic Survey Tele-
scope (LSST; http://www.lsst.org/lsst home.html) (Tyson (2002)). Similarly, there are many ongoing, planned and proposed
SNe Ia surveys, such as the Supernova Legacy Survey (Pain et al. (2002); Prichet (2004)) (SNLS); The Nearby Super-
nova Factory (SNfactory) (Wood-Vasey et al. (2004)); the ESSENCE project (Smith et al. (2002); Garnavich et al (2002);
Kirshner et al. (2003)); Sloan Digital Sky Survey (SDSS) (Madgwick et al. (2003)); The Carnegie Supernova Project
(Freedman et al. (2004)); and the Dark Energy Camera Project (DeCamera (2004)). We should mention that there are
other noteworthy cosmological tools for probing dark energy that we did not consider in this study, notably clusters of
galaxies (see for example (Mohr (2004); Wang et al.(2003)) and references therein), Lyman-alpha forests (see for example
(Mandelbaum et al (2003); Seljak et al. (2004))), and baryonic oscillations (see for example (Eisenstein (2003); Seo & Eisenstein (2003);
Blake & Glazebrook (2003); Linder (2003))).
For the CMB, we consider future constraints from 8 years of data from the Wilkinson Microwave Anisotropy Probe
(WMAP-8) (Bennet et al. (2003); Spergel et al. (2003)), 1 year of data from the Planck satellite (PLANCK1), and 1 year of
data from the Atacama Cosmology Telescope (ACT), see e.g. (Kosowsky (2003)). We use two sets of 2000 supernovae with
zmax =0.8 and zmax =1.5 respectively, and consider two types of cosmic shear surveys: a ground-based like survey covering
70% of the sky with source galaxy redshift distribution having a median redshift zmed ≈ 1, and a space-based like deep survey
covering successively 1%, 10% and 70% of the sky with zmed ≈ 1.5.
We take into account in our analysis systematic limits for the supernovae by adding a systematic uncertainty floor
in quadrature following (Kim et al. (2003)). We also include for weak lensing the effect of the redshift bias and the shear
calibration bias by adding and marginalizing over the corresponding parameters as in (Ishak et al. (2004)).
The constraints on the dark energy equation of state are parameterization dependent; see, e.g. (Wang & Tegmark (2004);
Upadhye et al. (2004)). Also, there is a smearing effect due to double integration involved when using the equation of state
(Maor et al. (2002); Maor et al. (2001)). In order to partly avoid this, one could probe directly the variations in the dark
energy density using the data; see, e.g. (Wang & Mukherjee (2004); Wang & Freese (2004)). However, it has been argued that
the equation of state is closer to the physics, as it also contains information on the pressure, and, trying to probe the equation
c 0000 RAS, MNRAS 000, 000–000
Probing decisive answers to dark energy questions from
cosmic complementarity and lensing tomography
Mustapha Ishak
Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544, USA
arXiv:astro-ph/0501594v2 15 Jul 2005
14 January 2014
ABSTRACT
We study future constraints on dark energy parameters determined from several com-
binations of cosmic microwave background experiments, supernova data, and cosmic
shear surveys with and without tomography. In this analysis, we look in particular for
combinations of experiments that will bring the uncertainties to a level of precision
tight enough (a few percent) to answer decisively some of the dark energy questions.
In view of the parameterization dependence problems, we probe the dark energy using
two variants of its equation of state w(z), and its energy density ρde (z). For the latter,
we model ρde (z) as a continuous function interpolated using dimensionless parameters
Ei (zi ) ≡ ρde (zi )/ρde (0). We consider a large set of 13 cosmological and systematic
parameters, and assume reasonable priors on the lensing and supernova systematics.
For CMB, we consider future constraints from 8 years of data from WMAP, one year
of data from Planck, and one year of data from the Atacama Cosmology Telescope
(ACT). We use two sets of 2000 supernovae with zmax = 0.8 and 1.5 respectively,
and consider various cosmic shear reference surveys: a wide ground-based like survey,
covering 70% of the sky, and with successively 2 and 5 tomographic bins; a deep space-
based like survey with 10 tomographic bins and various sky coverages. The one sigma
constraints found are {σ(w0 ) = 0.086, σ(w1 ) = 0.069}, {σ(w0 ) = 0.088, σ(wa ) = 0.11},
and {σ(E1 ) = 0.029, σ(E2 ) = 0.065} from Planck, supernovae and the ground-based
like lensing survey with 2 bins. When 5 bins are used within the same combination the
constraints reduce to {σ(w0 ) = 0.04, σ(w1 ) = 0.034}, {σ(w0 ) = 0.041, σ(wa ) = 0.056},
and {σ(E1 ) = 0.012, σ(E2 ) = 0.049}. Finally, when the deep lensing survey with
10% coverage of the sky and 10 tomographic bins is used along with Planck and the
deep supernovae survey, the constraints reduce to {σ(w0 ) = 0.032, σ(w1 ) = 0.027},
{σ(w0 ) = 0.033, σ(wa ) = 0.04}, and {σ(E1 ) = 0.01, σ(E2 ) = 0.04}. Other coverages of
the sky and other combinations of experiments are explored as well. Although some
worries remain about other systematics, our study shows that after the combination
of the three probes, lensing tomography with many redshift bins and large coverages
of the sky has the potential to add key improvements to the dark energy parameter
constraints. However, the requirement for very ambitious and sophisticated surveys in
order to achieve some of these constraints or to improve them suggests the need for
new tests to probe the nature of dark energy in addition to constraining its equation
of state.
Key words: cosmology: theory – dark energy – gravitational lensing – large-scale
structure of universe
1 INTRODUCTION
One of the most important and challenging questions in cosmology and particle physics is to understand the nature of the
dark energy that is driving the observed cosmic acceleration, see e.g. (Weinberg (1989); Carroll et al. (1992); Turner (2000);
Sahni & Starobinsky (2000); Padmanabhan (2003); Ishak (2005)). An important approach to this problem is to constrain
c 0000 RAS
, 2 M. Ishak
the properties of dark energy using cosmological probes. This would provide measurements that would allow one to test
various competing models of dark energy. However, due to a high degeneracy within a narrow range of the parameter space,
constraining conclusively dynamical dark energy models is going to be a difficult goal to achieve, and much effort and strategy
will be needed. Ultimately, a combination of powerful cosmological probes and tests will be necessary.
In this paper, we study how dark energy parameters are constrained from different combinations of cosmic microwave
background (CMB) experiments, supernovae of type Ia (SNe Ia) data, and weak lensing surveys (WL) with and without
tomography. In particular, we look for combinations of experiments that will be able to constrain these parameters well
enough to settle decisively some of the dark energy questions, say to a few percent. When CMB measurements constrained
the total energy density to ΩT = 1.02 ± 0.02 to a one sigma level (Spergel et al. (2003); Bennet et al. (2003)), it became
generally more accepted that spatial curvature is negligible. Thus, an uncertainty of a few percent on dark energy parameters
could be set as a reasonable goal. Of course, one should bear in mind that it will remain always possible to construct dynamical
dark energy models that are indistinguishable from a cosmological constant within these limits, and therefore, one needs to
resort to cosmological tests beyond the equation of state measurements. A better scenario providing a decisive answer would
be one in which one could show that dark energy is clearly not a cosmological constant.
It is certainly wise to probe the nature of dark energy using gradual steps. However, in both the scenarios mentioned
above, one should keep in mind that the results and conclusions obtained from an analysis where the equation of state is
assumed constant are subject to changes if the equation of state is allowed to vary with the redshift. In this paper, we consider
dark energy with a varying equation of state.
We chose the combination CMB+SN Ia+WL as various studies have already shown that supernovae type Ia constitute a
powerful probe of dark energy via distance-redshift measurements, see for example (Riess et al (1998); Garnavich et al. (1998);
Filippenko & Riess (1998); Perlmutter, et al. (1999); Perlmutter S., et al. (1997); Riess et al (2000); Riess et al (2001))
(Tonry et al. (2003); Knop et al. (2003) ; Barris et al,, (2004); Riess et al. (2004)). Also, several parameter forecast studies
have shown that combining constraints from weak gravitational lensing with constraints from the CMB is a powerful combina-
tion to constrain dark energy; see, e.g. (Hu (2001); Huterer (2002); Huterer & Turner (2001); Benabed & Van Waerbeke (2003))
(Abazajian & Dodelson (2003); Refregier et al. (2003); Heavens (2003) ; Simon et al (2003); Jain & Taylor (2003))
(Bernstein & Jain (2004); Song & Knox (2004)). Importantly, weak lensing measurements are sensitive to both the effect of
dark energy on the expansion history and its effect on the growth factor of large-scale structure. Furthermore, in addition
to tightening the constraints, using independent probes will allow one to test the systematic errors of each probe, which are
serious limiting factors in these studies. For each of these probes, much data will be available in the near and far future.
For WL, there are many ongoing, planned and proposed surveys, such as the Deep Lens Survey (http://dls.bell-labs.com/)
(Wittman et al.(2002)); the NOAO Deep Survey (http://www.noao.edu/noao/noaodeep/); the Canada-France-Hawaii Tele-
scope (CFHT) Legacy Survey (http://www.cfht.hawaii.edu/Science/CFHLS/) (Mellier et al.(2001)); the Panoramic Sur-
vey Telescope and Rapid Response System (http://pan-starrs.ifa.hawaii.edu/); the Supernova Acceleration Probe (SNAP;
http://snap.lbl.gov/) (Rhodes et al.(2003); Massey et al.(2003); Refregier et al.(2003)); and the Large Synoptic Survey Tele-
scope (LSST; http://www.lsst.org/lsst home.html) (Tyson (2002)). Similarly, there are many ongoing, planned and proposed
SNe Ia surveys, such as the Supernova Legacy Survey (Pain et al. (2002); Prichet (2004)) (SNLS); The Nearby Super-
nova Factory (SNfactory) (Wood-Vasey et al. (2004)); the ESSENCE project (Smith et al. (2002); Garnavich et al (2002);
Kirshner et al. (2003)); Sloan Digital Sky Survey (SDSS) (Madgwick et al. (2003)); The Carnegie Supernova Project
(Freedman et al. (2004)); and the Dark Energy Camera Project (DeCamera (2004)). We should mention that there are
other noteworthy cosmological tools for probing dark energy that we did not consider in this study, notably clusters of
galaxies (see for example (Mohr (2004); Wang et al.(2003)) and references therein), Lyman-alpha forests (see for example
(Mandelbaum et al (2003); Seljak et al. (2004))), and baryonic oscillations (see for example (Eisenstein (2003); Seo & Eisenstein (2003);
Blake & Glazebrook (2003); Linder (2003))).
For the CMB, we consider future constraints from 8 years of data from the Wilkinson Microwave Anisotropy Probe
(WMAP-8) (Bennet et al. (2003); Spergel et al. (2003)), 1 year of data from the Planck satellite (PLANCK1), and 1 year of
data from the Atacama Cosmology Telescope (ACT), see e.g. (Kosowsky (2003)). We use two sets of 2000 supernovae with
zmax =0.8 and zmax =1.5 respectively, and consider two types of cosmic shear surveys: a ground-based like survey covering
70% of the sky with source galaxy redshift distribution having a median redshift zmed ≈ 1, and a space-based like deep survey
covering successively 1%, 10% and 70% of the sky with zmed ≈ 1.5.
We take into account in our analysis systematic limits for the supernovae by adding a systematic uncertainty floor
in quadrature following (Kim et al. (2003)). We also include for weak lensing the effect of the redshift bias and the shear
calibration bias by adding and marginalizing over the corresponding parameters as in (Ishak et al. (2004)).
The constraints on the dark energy equation of state are parameterization dependent; see, e.g. (Wang & Tegmark (2004);
Upadhye et al. (2004)). Also, there is a smearing effect due to double integration involved when using the equation of state
(Maor et al. (2002); Maor et al. (2001)). In order to partly avoid this, one could probe directly the variations in the dark
energy density using the data; see, e.g. (Wang & Mukherjee (2004); Wang & Freese (2004)). However, it has been argued that
the equation of state is closer to the physics, as it also contains information on the pressure, and, trying to probe the equation
c 0000 RAS, MNRAS 000, 000–000
