Cosmology with High-redshift Galaxy Survey: Neutrino Mass and Inflation Masahiro Takada1, Eiichiro Komatsu2 and Toshifumi Futamase1 1Astronomical Institute, Tohoku University, Sendai 980-8578, Japan and 2Department of Astronomy, The University of Texas at Austin, Austin, TX 78712 High-z galaxy redshift surveys open up exciting possibilities for precision determinations of neu- trino masses and inflationary models. The high-z surveys are more useful for cosmology than low-z ones owing to much weaker non-linearities in matter clustering, redshift-space distortion and galaxy bias, which allows us to use the galaxy power spectrum down to the smaller spatial scales that are inaccessible by low-z surveys. We can then utilize the two-dimensional information of the linear power spectrum in angular and redshift space to measure the scale-dependent suppression of matter clustering due to neutrino free-streaming as well as the shape of the primordial power spectrum. To illustrate capabilities of high-z surveys for constraining neutrino masses and the primordial power spectrum, we compare three future redshift surveys covering 300 square degrees at 0:5 < z < 2, 2 < z < 4, and 3:5 < z < 6:5. We find that, combined with the cosmic microwave background data expected from the Planck satellite, these surveys allow precision determination of the total neutrino mass with the projected errors of σ(mν,tot) = 0:059, 0.043, and 0.025 eV, respectively, thus yielding a positive detection of the neutrino mass rather than an upper limit, as σ(mν,tot) is smaller than the lower limits to the neutrino masses implied from the neutrino oscillation experiments, by up to a factor of 4 for the highest redshift survey. The accuracies of constraining the tilt and running index −3 −3 of the primordial power spectrum, σ(ns) = (3:8; 3:7; 3:0) × 10 and σ(αs) = (5:9; 5:7; 2:4) × 10 −1 at k0 = 0:05 Mpc , respectively, are smaller than the current uncertainties by more than an or- der of magnitude, which will allow us to discriminate between candidate inflationary models. In particular, the error on αs from the future highest redshift survey is not very far away from the prediction of a class of simple inflationary models driven by a massive scalar field with self-coupling, −3 αs = −(0:8 − 1:2) × 10 . PACS numbers: 95.55.Vj,98.65.Dx,98.80.Cq,98.70.Vc,98.80.Es I. INTRODUCTION perature anisotropy [10], temperature{polarization cross correlation [11], and distribution of galaxies [12, 13]. We are living in the golden age of cosmology. Vari- Yet, the latest observations have shown convincingly ous data sets from precision measurements of tempera- that we still do not understand much of the universe. The ture and polarization anisotropy in the cosmic microwave standard model of cosmology tells us that the universe background (CMB) radiation as well as those of matter has been dominated by four components. In chronolog- density fluctuations in the large-scale structure of the ical order the four components are: early dark energy universe mapped by galaxy redshift surveys, Lyman-α (also known as “inflaton” fields), radiation, dark mat- forests and weak gravitational lensing observations are ter, and late-time dark energy. The striking fact is that in a spectacular agreement with the concordance ΛCDM we do not understand the precise nature of three (dark model [1{4]. These results assure that theory of cos- matter, and early and late-time dark energy) out of the mological linear perturbations is basically correct, and four components; thus, understanding the nature of these can accurately describe the evolution of photons, neu- three dark components has been and will continue to be trinos, baryons, and collisionless dark matter particles one of the most important topics in cosmology in next [5{7], for given initial perturbations generated during in- decades. Of which, one might be hopeful that the next flation [8, 9]. The predictions from linear perturbation generation particle accelerators such as the Large Hadron theory can be compared with the precision cosmological Collider (coming on-line in 2007) would find some hints measurements, in order to derive stringent constraints on for the nature of dark matter particles. On the other the various basic cosmological parameters. Future obser- hand, the nature of late-time dark energy, which was dis- vations with better sensitivity and higher precision will covered by measurements of luminosity distance out to continue to further improve our understanding of the uni- distant Type Ia supernovae [14, 15], is a complete mys- verse. tery, and many people have been trying to find a way to Fluctuations in different cosmic fluids (dark matter, constrain properties of dark energy (see, e.g., [16] for a photons, baryons, and neutrinos) imprint characteristic review). features in their power spectra, owing to their interac- How about the early dark energy, inflaton fields, which tion properties, thermal history, equation of state, and caused the expansion of the universe to accelerate in the speed of sound. A remarkable example is the acoustic very early universe? We know little about the nature oscillation in the photon-baryon fluid that was generated of inflaton, just like we know little about the nature of before the decoupling epoch of photons, z 1088, which late-time dark energy. The required property of infla- has been observed in the power spectrum'of CMB tem- ton fields is basically the same as that of the late-time 2 dark energy component: both must have a large negative physics: the cosmological observations will provide inde- pressure which is less than 1=3 of their energy density. pendent tests of this hypothesis. − To proceed further, however, one needs more information In this paper we shall study the capability of future from observations. Different inflation models make spe- galaxy surveys at high redshifts, combined with the CMB cific predictions for the shape of the power spectrum [8] data, for constraining (1) the neutrino properties, more (see also Appendix B) as well as for other statistical prop- specifically the total neutrino mass, mν,tot, and the num- nr erties [17] of primordial perturbations. Therefore, one of ber of non-relativistic neutrino species, Nν , and (2) the the most promising ways to constrain the physics of in- shape of the primordial power spectrum that is parame- flation, hence the nature of early dark energy in the uni- terized in terms of the spectral tilt, ns, and the running verse, is to determine the shape of the primordial power index, αs, motivated by inflationary predictions (see Ap- spectrum accurately from observations. For example, the pendix B). For the former, we shall pay particular at- CMB data from the Wilkinson Microwave Anisotropy tention to our ability to simultaneously constrain mν,tot nr Probe [1], combined with the large-scale structure data and Nν , as they will provide important clues to resolv- from the Two-Degree Field Galaxy Redshift Survey [18], ing the absolute mass scale as well as the neutrino mass have already ruled out one of the popular inflationary hierarchy. The accuracy of determining the neutrino pa- models driven by a self-interacting massless scalar field rameters and the power spectrum shape parameters will [19]. Understanding the physics of inflation better will be derived using the Fisher information matrix formal- likely provide an important implication for late-time dark ism, including marginalization over the other cosmologi- energy. cal parameters as well as the galaxy bias. \Radiation" in the universe at around the matter- Our analysis differs from the previous work on the radiation equality mainly consists of photons and neu- neutrino parameters in that we fully take into account trinos; however, neutrinos actually stop being radiation the two-dimensional nature of the galaxy power spec- when their mean energy per particle roughly equals the trum in the line-of-sight and transverse directions, while temperature of the universe. The physics of neutrinos the previous work used only spherically averaged, one- has been revolutionized over the last decade by solar, dimensional power spectra. The geometrical distortion atmospheric, reactor, and accelerator neutrino experi- due to cosmology and the redshift space distortion due to ments having provided strong evidence for finite neu- the peculiar velocity field will cause anisotropic features trino masses via mixing between different neutrino fla- in the galaxy power spectrum. These features help to vors, the so-called neutrino oscillations [20{24]. These lift degeneracies between cosmological parameters, sub- experiments are, however, only sensitive to mass square stantially reducing the uncertainties in the parameter de- differences between neutrino mass eigenstates, implying terminations. This is especially true when variations in 2 −5 2 2 −3 2 parameters of interest cause modifications in the power ∆m21 7 10 eV and ∆m32 3 10 eV ; thus, the most' fundamen× tal quantity of'neutrinos,× the abso- spectrum shape, which is indeed the case for the neu- lute mass, has not been determined yet. Cosmological trino parameters, tilt and running index. The useful- neutrinos that are the relic of the cosmic thermal his- ness of the two-dimensional power spectrum, especially tory have distinct influences on the structure formation. for high-redshift galaxy surveys, has been carefully in- Their large energy density, comparable to the energy den- vestigated in the context of the prospected constraints sity of photons before the matter-radiation equality, de- on late-time dark energy properties [39{45]. termines the expansion history of the universe. Even We shall show the parameter forecasts for future wide- after the matter-radiation equality, neutrinos having be- field galaxy surveys that are already being planned or come non-relativistic affect the structure formation by seriously under consideration: the Fiber Multiple Object suppressing the growth of matter density fluctuations at Spectrograph (FMOS) on Subaru telescope [46], its sig- small spatial scales owing to their large velocity disper- nificantly expanded version, WFMOS [47], the Hobby{ sion [25, 26, 29, 30] (see Sec.