The Dark Energy Spectrometer (despec): a multi-Fiber Spectroscopic Upgrade of the Dark Energy Camera and Survey for the Blanco Telescope

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We describe a new initiative to build and use the Dark Energy Spectrometer (DESpec), a wide-field, deep spectroscopic survey instrument for the Blanco 4-meter telescope at Cerro Tololo Inter-American Observatory (CTIO) in Chile. A new system with about 4000 robotically positioned optical fibers will be rapidly interchangeable with the CCD imager of the existing Dark Energy Camera (DECam), accessing a field of view of 3.8 square degrees in a single exposure. The proposed? instrument will be operated by CTIO and available for use by the astronomy community. Our collaboration proposes to conduct a spectroscopic survey to study Dark Energy, sharing the survey products with the community.

In a survey of ~300 nights, the DESpec collaboration proposes to obtain spectroscopic redshifts for ~7 million galaxies over 5000 sq. deg. selected from precision imaging from the Dark Energy Survey (DES). This Dark Energy Spectroscopic Survey will advance our knowledge of cosmic expansion and structure growth significantly beyond that obtainable with imaging-only surveys. Since it adds a spectroscopic third dimension to the same sky as DES, the world’s largest and deepest photometric survey, DESpec will enable a powerful array of new increasingly precise techniques to discriminate among alternative explanations of cosmic acceleration, such as Dark Energy and Modified Gravity. In the 2020 decade, a wider-area spectroscopic survey will enhance the science reach of the next-generation imaging project, LSST, by providing spectroscopy for tens of millions of galaxies and QSO’s over its 15,000-20,000 square degree survey area. DESpec will create a comprehensive, three dimensional, critically-sampled map of large scale cosmic structure a database encompassing the “cosmic web” of galaxies and clusters over the entire, deeply-imaged southern sky over most of cosmic history.

DESpec will take advantage of the substantial hardware infrastructure recently built for DECam to achieve excellent science at low cost and low technical and schedule risk. It will be mounted in the DECam prime focus cage with its hexapod positioning system (recently installed at the Blanco Telescope) and will share the four largest DECam optical corrector elements. It will be routinely interchangeable with the DECam CCD imager and preserve f/8 secondary capability to allow flexible use of the telescope. The robotically positioned optical fibers will feed 10 relatively simple, high-throughput spectrometers outfitted with existing, spare, red-sensitive, science-grade DECam CCDs. An Atmospheric Dispersion Compensator similar to existing designs is accommodated in the existing DECam filter slot. Examples that meet requirements for all major subsystems are in existing or highly developed instruments, showing that cost, technical and schedule risks are manageable.

1. Introduction

1.A. The Foundational Impact of a Wide, Deep Spectroscopic Survey

In 1998, two teams of astronomers studying distant Type Ia supernovae presented evidence that the expansion of the Universe is speeding up rather than slowing down due to gravity (Riess, et al. 1998, Perlmutter, et al. 1999). In the following years, maps of cosmic structure produced by the Sloan Digital Sky Survey (SDSS), and deep maps of the cosmic microwave background by Boomerang, MAXIMA, DASI, WMAP and other experiments, confirmed and extended this result, consolidating for the first time a precise, comprehensive model for the overall size, shape, and contents of the universe. These transformational discoveries were recognized as Science magazine’s “Breakthroughs of the Year” in 1998 and 2003. In 2011, Saul Perlmutter, Adam Riess, and Brian Schmidt were awarded the Nobel Prize in Physics for the discovery of cosmic acceleration.

While cosmology is now on a much more precise footing than before, the physical origin of cosmic acceleration remains a mystery (e.g., Frieman, Turner, & Huterer 2008; Sullivan, et al. 2011). Unraveling it will have profound implications for fundamental physics. Is acceleration caused by an exotic new form of Dark Energy (DE) that makes up most of the Universe? Or, does it indicate that Einstein’s Theory of General Relativity (GR) must be replaced by a new theory of gravity on cosmic scales? If the answer is dark energy, is it the energy of the vacuum Einstein’s “cosmological constant” or something else, perhaps an ultralight scalar field, dubbed quintessence? Some physicists even speculate that cosmic acceleration is a sign of new emergent physics that transcends our current separate notions of mass-energy and space-time.

This enduring and profound mystery can be addressed with better measurements of the history of the cosmic expansion rate and of the growth of large-scale structure. Ambitious next steps are already underway: the Dark Energy Survey (DES), starting later this year, and the Large Synoptic Survey Telescope (LSST) in the 2020 decade, will extend wide field imaging well beyond the depth of SDSS. They will create precision images of the sky with hundreds of millions to billions of galaxies, deep and broad enough to probe the history of cosmic acceleration and structure formation in unprecedented detail. Like SDSS, their databases of the deep sky will lead to a wide range of new probes and discoveries.

While the SDSS combined imaging and spectroscopy in a single survey, DES and LSST are imaging-only projects: they do not yet include a comprehensive spectroscopic survey. Detailed spectra, with resolution of several thousand, were important contributors to many discoveries with SDSS: they provide precision redshifts and distances and a wealth of information about the light-emitting sources as well as absorbing matter along the line of sight. Spectroscopic information will be even more critical in future studies, as we seek higher precision probes and better control over many unforeseen systematic errors.

In this document, we describe the Dark Energy Spectrometer, a low-risk, cost-effective way to create a wide, deep spectroscopic survey to accompany the DES and LSST imaging surveys. The system uses many existing components of the Dark Energy Camera, such as the wide-field corrector optics. The DECam CCD imager is rapidly removed and replaced by a new robotic fiber system that measures simultaneous spectra of about 4000 galaxies in a field of 3.8 square degrees.

In just two 30-minute exposures, this system critically samples all large-scale spatial structure in our past light cone out to a redshift of about 1.5, as traced by galaxies with mean separation ~15 h-1Mpc. Their redshifts provide a high-resolution, comprehensive, three-dimensional map of the cosmic web extending over most of its history from here and now back to before large-scale structure formed and before the cosmic expansion started to accelerate.

Using DES imaging to select targets, DESpec will cover about 5000 square degrees with this depth and sampling in just a few hundred nights. We show below how this large survey, with about 7 million spectra of DES sources, will provide new and uniquely powerful probes of dark energy physics.

This large initial survey could be in place at or near the beginning of the LSST survey. The precise map of large-scale structure would give a powerful boost to a wide range of early LSST science. After several more years, using LSST to select additional targets, the spectroscopic survey will be extended to the entire southern sky and perhaps as many as 30 million sources.

The rest of this white paper emphasizes the value of the DESpec survey for understanding the physics of cosmic acceleration in ways we can calculate today and provides initial concepts for the DESpec design. We note however that as with SDSS, this massive and versatile spectroscopic database will certainly amplify the science impact of DES and LSST over a much broader range of enquiry in many ways we cannot yet foresee, and that may indeed lead to more profound discoveries.

1.B. DESpec: the Dark Energy Spectrometer and Spectroscopic Survey

The Dark Energy Survey (DES) is a deep, wide, multi-band imaging survey, spanning 525 nights over five years beginning in late 2012, that will use the new 570-Megapixel Dark Energy Camera on the Blanco 4-m telescope at CTIO (DES Collaboration, 2005). DES, in partnership with ESO’s near-infrared VISTA Hemisphere Survey (VHS), will provide imaging of ~300 million galaxies in 5+3 optical-NIR filters (grizY for DES, JHK for VHS) over 5000 sq. deg. DES will also discover and measure ~4000 SN Ia light curves in a time-domain survey of 30 sq. deg. With this survey, DES will probe dark energy using four techniques: the clustering of galaxies on large scales, including baryon acoustic oscillations (BAO); the abundance of massive galaxy clusters; weak gravitational lensing distortions of the images of distant galaxies; and Type Ia supernova distances. DES is an international collaboration, with over 130 senior scientists from 27 institutions in the US, the UK, Spain, Brazil, Germany, and Switzerland. Funding for DES within the US is provided by the Department of Energy (DOE), the National Science Foundation (NSF), and the participating US institutions. DES is partially supported by the following foreign science agencies and programs: the UK Science and Technology Facilities Council (formerly PPARC); the UK Science Research Infrastructure Fund (SRIF3); the Astronomy and Astrophysics program of the Ministry of Science and Innovation (MICINN) of the Spanish Government; the Brazilian Government agency FINEp; the Brazilian Ministry of Science & Technology (MCT); the Brazilian state science agency FAPERJ and the federal agency CNPq; and the German science foundation’s Excellence Cluster “Origin and Structure of the Universe.”
In the language of the Dark Energy Task Force report (DETF, Albrecht, et al. 2006), DES is a ‘Stage III’ experiment that will make a substantial step forward in constraining the properties of dark energy.

As a multi-band imaging survey, DES (and later LSST) will provide precise measurements of galaxy fluxes, colors, and shapes but only approximate photometric estimates of their redshifts (photo-z’s). High-precision redshifts, which enable a true 3d map of the cosmos, require spectroscopy. The DESpec collaboration seeks to substantially enhance the science reach of DES imaging, and better probe the origin of cosmic acceleration, by obtaining spectroscopic redshifts for a large sample of DES target galaxies, yielding a dense sampling of 3d structure over the wide, deep volume probed by DES.

The need for such data motivates DESpec, a concept for a ~4000-fiber spectrograph for the Blanco telescope, which enables a ~7 million spectroscopic galaxy survey in ~300 nights. We consider DESpec an ‘upgrade’ of the Dark Energy Camera and of the DES project, following the model of upgrades that enhance the capabilities of high-energy particle physics experiments. Together with DES, DESpec has the science reach of a next-generation, DETF Stage IV project. Once LSST begins survey operations from neighboring Cerro Pachon, the DESpec survey can be expanded to ~15,000 sq. deg., yielding redshifts for ~20 million galaxies in ~900 nights, substantially enhancing the science reach of LSST. This sample is comparable in size to, and complements, the proposed BigBOSS survey in the north (Schlegel, et al. 2011; see Appendix).

However, the unique strength of the DESpec survey transcends the statistics captured in the DETF figure of merit: it will provide a 3d redshift map of the Universe over the same deep, wide volume precisely mapped by DES and later LSST. The statistical versatility and power of a comprehensive, deep 3d survey coupled with a precision, multi-band, homogeneous photometric survey was demonstrated by SDSS, which was instrumental in establishing the current cosmological paradigm. To probe the physical origin of cosmic acceleration will require a combination of many statistical techniques, only some of which are now known and well tested. DESpec will survey a volume much larger than SDSS, extending back to an era before Dark Energy dominated the cosmic expansion. DESpec enables a wide range of DE probes that powerfully synergize in ways that no other foreseen spectroscopic surveys can achieve.

The spectroscopic redshift information that DESpec adds to the DES(+VHS) galaxy catalog results in a substantial increase in the precision of the dark energy equation of state parameter, w, and its time evolution, dw/da, from all four of the techniques above (baryon acoustic oscillations; abundance of massive galaxy clusters; weak gravitational lensing; Type Ia supernova distances). Spectroscopic redshift precision also enables qualitatively new dark energy probes beyond DES, for example, radial baryon acoustic oscillations and redshift-space distortions (RSD). It can provide dynamical mass estimates for thousands of DES galaxy clusters, strengthening the cluster probe of DE. Spectroscopic data also increase the power of weak lensing as a probe in controlling intrinsic alignment effects. In the southern hemisphere, this includes cross-correlation with lensing of the microwave background radiation, now detected in high resolution imaging by the South Pole Telescope and the Atacama Cosmology Telescope.

Among the many new capabilities enabled by DESpec, one exciting recent realization is that the combination of redshift-space distortions from DESpec and weak lensing from DES (and later LSST) can powerfully discriminate models of Modified Gravity from Dark Energy as the cause of cosmic acceleration (Zhang, et al. 2007, Guzik, et al. 2009, Song & Dore 2009, Reyes, et al. 2010, Song, et al. 2011), with reduced uncertainty due to galaxy bias and cosmic variance if the photometric and spectroscopic surveys cover the same sky area as DES and DESpec would do (Pen 2004, McDonald & Seljak 2009, Cai & Bernstein 2012, Gaztanaga, et al. 2012). In this White Paper we provide further analysis of combining imaging and spectroscopic data over the same sky, and outline our R&D program to study it further by three teams within the DESpec collaboration. Qualitatively our three independent studies indicate that same-sky for imaging and spectroscopy is superior to non-overlapping photometric and spectroscopic surveys.

Further, the DESpec survey can set a strict upper limit on or even detect the mass of neutrinos, with expected sensitivity of ~0.05 eV in combination with DES and Planck, highly competitive with laboratory experiments. This very large spectroscopic survey will also probe the presently puzzling excess power in the galaxy clustering power spectrum on the Gigaparsec scale (Thomas, Abdalla, & Lahav 2011) and enable new studies of galaxy evolution and of quasars. Additional payoffs include reducing DES weak lensing intrinsic alignment systematics by cross-correlating the lensing shear signal with a spectroscopic galaxy sample, improved determination of the redshift distribution of DES photometric galaxies via angular cross-correlation with a spectroscopic sample, thereby reducing systematic errors of all of the Dark Energy probes, detailed study of the dark matter environments of galaxies and clusters via stacked weak lensing mass estimates, and measurement of cluster dynamical masses via velocity dispersions. These arguments are strengthened when extended to wider-area follow-up of LSST imaging.

A variety of factors combine to argue for a deep spectroscopic survey from the southern hemisphere. A well calibrated, uniform, deep target list of objects with precision multi-band photometry and lensing shape measurements is guaranteed from DES. DESpec can access the entire sky area that will be surveyed by both DES (5000 sq. deg.) and later LSST (20,000 sq. deg.), as well as the highest resolution mappers of the cosmic microwave background, the South Pole Telescope and the Atacama Cosmology Telescope. Located in the southern hemisphere at one of the world’s premier astronomical sites (CTIO has a median site seeing of 0.65” FWHM and 80% useable nights), DESpec will yield a wealth of spectroscopic survey information complementary to the larger-aperture, narrower-field VLT, Gemini, Magellan, and other telescopes concentrated in the southern hemisphere. In the farther future, southern sites are currently planned for two of three next-generation very large optical telescopes, as well as the Square Kilometer Array in the radio. A similar wide-field spectroscopic survey currently under development, the BigBOSS project on the Kitt Peak Mayall 4-meter telescope, is in the northern hemisphere. The Sumire Prime Focus Spectrograph, under development for the larger-aperture Subaru Telescope, has a smaller field of view and is also in the northern hemisphere. ESO’s 4MOST project has a wider field-of-view but is optimized for spectroscopic follow-up of stars astrometrically measured by Gaia.

In addition to its dark energy goals, as a community instrument DESpec will enable a wide array of coordinated spectroscopic surveys, affording opportunities for discoveries in stellar structure and evolution, nearby galaxies, galaxy evolution, the structure of galaxy clusters, and other applications. Given the large number of fibers, building on the experience of the Sloan Digital Sky Survey, one can optimize efficiency by conducting many survey programs in parallel. The ability to interchange the instrument with DECam enables a flexible observing program including wide-field imaging in the pre-LSST era. This program efficiently exploits the unique wide field and structural capabilities of the Blanco telescope. The DESpec/DECam system creates an opportunity for optimal use of resources by both DOE and NSF user communities: a large science impact per dollar and a beneficial partnership combining the technical resources of both agencies. In the LSST era, the wide-field spectroscopic capability of DESpec complements two other capabilities one following up very faint LSST sources over small sky areas (e.g., with ~8 to 30-meter telescopes and with JWST) and the other following up optical LSST transients on very rapid timescales to form an optimized southern spectroscopic system.

DESpec achieves relatively low cost, schedule, and technical risk by capitalizing on and leveraging the recent investment in DECam and recent structural improvements in the Blanco telescope and its environment made by NOAO (new primary mirror radial supports, improved telescope control system, environmental control, and other infrastructure). It uses the DECam mechanical structure (prime focus cage, barrel, shutter, hexapod alignment system), four of the same corrector lenses, and an ample supply (~60) of existing spare, packaged and tested, red-sensitive, science-grade 2kx4k CCDs produced for the DECam project. Two new optical corrector lenses will be needed, with the need for an Atmospheric Dispersion Compensator still under review. The resulting optical beam is nearly telecentric. The preliminary reference design described here entails ~4000 robotically positioned optical fibers of diameter 1.75” (100 microns) over the existing DECam 3.8 sq. deg. field of view, feeding 10 double-arm spectrographs. We are also investigating the science reach and cost trade-offs of a single-arm design for the spectrographs.

One low-cost spectrograph design builds upon that used for the Hobby-Eberly Telescope Dark Energy Experiment (HETDEX). The CCD readout electronics will be copied from the low-read-noise DECam design, again saving cost and reducing risk. The two-arm (dichroic) spectrograph would have a blue side wavelength range 480<<780 nm, and the red side would cover 750<<1050 nm. Over this range, the optical corrector produces an appropriately small spot size of 0.4-0.63” FWHM. These design numbers fiber size, wavelength range, spectral resolution, and resulting spectrograph design will be optimized in the R&D phase. Since NOAO expects to operate DECam for at least five years beyond the end of DES (i.e., to 2022), DESpec will be designed to be interchangeable with DECam. The design preserves the Blanco’s capability to support an f/8 secondary, allowing flexible use of the telescope with other instruments at the Cassegrain focus.

An R&D program for DESpec now underway focuses on: (1) optimizing the optical and spectrograph design, balancing science requirements against cost and schedule and demonstrating feasibility; (2) establishing requirements for and demonstrating technological feasibility of the fiber-positioning system, likely in coordination with the group developing the Echidna fiber-positioning system whose prototype appears to meet our requirements; (3) building upon DECam expertise to complete design of the CCD readout electronics and mechanical design. The R&D program will produce a final design and construction cost and schedule estimates.

The collaboration will seek project funding from the US agencies DOE and NSF (with major hardware funding to be proposed to the DOE), international funding agencies, and the participating institutions, following the successful model of the DES project. A number of DES and non-DES institutions will participate. There is substantial interest in the project from foreign institutions. The UK STFC recently awarded a £160k grant for DESpec R&D. The Australian Astronomical Observatory (AAO), a long-time world leader in astronomical fiber positioning systems and surveys, has indicated keen interest in taking a lead role in construction of the fiber positioning system.

We will propose that the instrument be operated by the National Optical Astronomy Observatory (NOAO), which operates CTIO, under a time allocation agreement with the collaboration to be negotiated (as was done for DES, which was allocated 525 nights in exchange for DECam). However, it is important to note that NOAO at this time has not yet issued nor indicated that they will issue an announcement of opportunity for a new prime focus instrument for the Blanco Telescope.

Section 2 of this White Paper outlines the science case for DESpec, focusing on probes of dark energy and tests of General Relativity. Section 3 discusses the selection of spectroscopic targets and defines a strawman survey strategy that follows from the science requirements of Section 2. Section 4 discusses the DESpec instrument components, highlighting those areas that would be addressed in the R&D phase. An appendix briefly compares DESpec with the proposed BigBOSS project.

2. DESpec Dark Energy Science Program

2.A. Probing the origin of cosmic acceleration and testing General Relativity

The Dark Energy Survey (DES) will enable measurements of the dark energy and dark matter densities and of the dark energy equation of state through four methods: galaxy clusters, weak gravitational lensing (WL), galaxy angular clustering (including angular Baryon Acoustic Oscillations, BAO), and supernovae (SNe). A spectroscopic redshift survey of a substantial fraction of DES target galaxies with an instrument such as DESpec would enhance each of these techniques, particularly the measurement of baryon acoustic oscillations (BAO) through galaxy clustering, and enable a new DE probe, redshift-space distortions (RSD). Moreover, the combination of RSD from DESpec with WL from DES and subsequently LSST would enable a powerful new test of the consistency of the General Relativity-plus-Dark Energy paradigm and therefore help distinguish Dark Energy from Modified Gravity as the physical cause of cosmic acceleration.

To frame the discussion, we outline the DES survey and a strawman concept for the DESpec survey the latter discussed in Sec. 3 and introduce figures of merit for DE and parameters for testing departures from General Relativity.

The DES will comprise two multi-band imaging surveys, a wide-field survey and a narrow time-domain survey. The wide-field survey will nominally cover 5000 sq. deg. in the south Galactic cap, all at high galactic latitude suitable for extragalactic studies, reaching ~24th magnitude in the grizY filters. The depth and filter coverage of the wide-field survey were chosen primarily to achieve accurate galaxy and cluster photo-z and shape measurements to redshifts z>1. The wide-field survey will detect over 100,000 galaxy clusters and measure shapes, photo-z’s, and positions for ~200 million galaxies. It will overlap completely with the ESO Vista Hemisphere Survey (VHS), which will obtain moderately deep imaging in J, H, and K filters. The combined 8-filter data will extend the range of precise galaxy photo-z’s to z~2. The DES Supernova Survey involves frequent (every few days) imaging of a 30 sq. deg. area in the griz filters, which will yield well-measured light curves for ~4000 Type Ia supernovae to redshifts z~1.

The DETF defined a figure of merit (FoM) for dark energy surveys by parametrizing the redshift evolution of the dark energy equation of state parameter by w(a)=w0+wa(1-a), where a(t)=1/(1+z) is the cosmic scale factor, w0 is the current value of w, and wa is a measure of its evolution with redshift. The DETF FoM is proportional to the reciprocal of the area in the w0-wa plane that encloses the 95% CL region. Defining a pivot epoch, ap, at which the uncertainty in w(a) is minimized for a given experiment, the DETF FoM is [(wp)(wa)]-1. More complex figures of merit have also been proposed (Albrecht & Bernstein 2007, Albrecht, et al. 2009). The DETF report provided an estimate for the Stage II FoM of about 60, where Stage II includes projections from surveys that were on-going at the time the report was written, combined with the forecast statistical precision of Planck CMB measurements on cosmological parameters. Using similar techniques, the DES collaboration estimated a combined FoM from all four techniques of about 260 for the final survey, characteristic of a DETF Stage III project. There is considerable uncertainty in this forecast, since there are large uncertainties in the ultimate levels of systematic errors for each of the techniques. According to the DETF, next-generation, Stage IV projects would be anticipated to increase the DETF FoM by another factor of ~3-5 compared to Stage III. Our initial projections indicate that the DES+DESpec combination would reach that level of precision.

In testing Modified Gravity (MG) vs. Dark Energy, it is also useful to have parameters and FoMs describing departures from General Relativity, and several have been proposed. In GR+DE, the linear growth rate of density perturbations (a) that form large-scale structure is uniquely determined by the expansion rate H(a) and the matter density parameter m. In particular, the logarithmic growth rate is given by f(a)=dln/dlna=m(a), where the growth exponent =0.55 in GR. In Modified Gravity theories, the relation between expansion history and perturbation growth can be changed; e.g., in the DGP braneworld model (Dvali, et al. 2000), =0.68. One FoM for modified gravity models is therefore [()]-2 (Albrecht et al. 2009). More general MG parameterizations have also been considered, as discussed below in Sec. 3.

In addition to dark energy and modified gravity, DES+DESpec would probe other areas of fundamental physics, including neutrinos and primordial non-Gaussianity from inflation. Current constraints on the sum of neutrino masses from large-scale structure depend somewhat on details and assumptions of the analyses, but a relatively conservative recent analysis finds an upper bound of 0.28 eV at 95% CL (Thomas, Abdalla, and Lahav 2010), and DES+Planck is expected to reach 0.1 eV (Lahav, et al. 2010). We estimate that DES+DESpec clustering measurements plus Planck would improve this bound to ~0.05 eV, reaching the regime where neutrino oscillation experiments indicate a detection is likely. Measurement of large-scale clustering in DES+DESpec, particularly constraints on the scale-dependence of galaxy bias, will constrain departures from primordial Gaussianity and thereby test models of primordial inflation.

For our baseline DESpec survey, we assume ~8 million successful redshifts are acquired over the 5000 sq. deg. DES footprint. By using flux, colors, and surface brightness to target a mixture of Luminous Red Galaxies (LRGs) at z<1 and Emission Line Galaxies (ELGs) at redshifts 0.6<z<1.7, we assume for purposes of illustration that the redshift distribution can be sculpted to be approximately constant over the redshift range 0.2<z<1.5. Sec. 3 describes how such a selection could be carried out. An extended DESpec survey would extend this targeting to ~24 million galaxies over 15,000 sq. deg. of extragalactic sky, by selecting targets from LSST. We note that, once it is operational, LSST will immediately reach the depth needed for selecting DESpec targets over its full survey area. We emphasize that this survey plan is just a strawman; the final target selection for DESpec (redshift distribution, flux limits, color selection, mix of LRG and ELG targets, exposure times) will follow from detailed R&D and science trade studies.

2.B Weak Lensing and Redshift Space Distortions

DESpec spectroscopy would enable measurement of galaxy clustering in redshift space. A galaxy’s redshift is the combination of its Hubble flow motion and the radial component of its peculiar velocity due to nearby structures. Clustering in redshift space is therefore distorted (anisotropic) relative to clustering in real space, due to the effects of peculiar velocities. In linear perturbation theory and assuming linear bias between galaxies and dark matter, the galaxy density perturbation Fourier amplitude in redshift space is given by (Kaiser 1987)

where b is the linear bias factor for the given population of galaxies, f is the linear growth factor defined above, is the cosine of the angle between k and the line of sight (not the same as in Sec. 2.A), and (k) is the real-space dark matter perturbation Fourier amplitude. Measurement of the anisotropy (-dependence) of the galaxy power spectrum in redshift space thus provides a measure of the growth rate of fluctuations, f(a), which in turn is sensitive to the properties of dark energy or modified gravity.

Figure 2.1 (from Gaztanaga, et al. 2011) shows a first estimate of the 68% CL statistical constraints on w0, wa, and  that would be expected with a baseline 5000 sq. deg. DESpec survey from redshift space distortions alone (RSD, purple), from DES weak lensing alone (Shear, blue), from the combination of RSD and weak lensing in non-coincident parts of the sky (red, RSD+Shear, e.g., from BigBOSS+DES assuming the baseline 5000 sq. deg. footprint of DES), and from the coincident combination of RSD+WL (yellow, i.e., DESpec+DES). It is found that the combination of weak lensing and RSD measurements in the same spatial volume (yellow), as DESpec plus DES would provide, leads to stronger constraints on dark energy and modified gravity (Cai & Bernstein 2012, Gaztanaga, et al. 2012).

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