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Geometrical Constraint on Curvature with BAO Experiments

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Geometrical Constraint on Curvature with BAO Experiments Geometrical constraint on curvature with BAO experiments Masahiro Takada1 and Olivier Dore´2;3 1Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU, WPI), The University of Tokyo Institutes for Advanced Study, The University of Tokyo, Chiba 277-8583, Japan 2Caltech M/C 350-17, Pasadena, CA 91125, USA 3Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, California, U.S.A. The spatial curvature (K or ΩK ) is one of the most fundamental parameters of an isotropic and homogeneous universe and has a close link to the physics of the early Universe. Combining the radial and angular diameter distances measured via the baryon acoustic oscillation (BAO) experiments allows us to unambiguously constrain the curvature. The method is primarily based on the metric theory, but is less sensitive to the theory of structure formation other than the existence of the BAO scale and is free of any model of dark energy. In this paper, we estimate a best achievable accuracy of constraining the curvature with the BAO experiments. We show that an all-sky, cosmic-variance-limited galaxy survey covering the Universe up to z ∼> 4 enables a precise determination −3 of the curvature to an accuracy of σ(ΩK ) ' 10 . When we assume a model of dark energy – either the −4 cosmological constant or the (w0; wa) model – it can achieve a precision of σ(ΩK ) ' a few × 10 . These forecasts require a high sampling density of galaxies, and are degraded by up to a factor of a few for a survey with a finite number density of ∼ 10−3(h=Mpc)3. PACS numbers: 98.80.-k,95.36.+x,95.75.-z,98.65.Dx I. INTRODUCTION the assumptions are relaxed – for instance, if a generalized model of dark energy is employed – the CMB constraint on the curvature is largely degraded [22]. The baryonic acoustic The curvature of the Universe (hereafter denoted as K or oscillation (BAO) provides us with an alternative, powerful Ω ) is one of the most fundamental quantities in an isotropic K geometrical probe, allowing one to constrain the cosmologi- and homogeneous universe in the context of general relativ- cal distances via measurements of the galaxy clustering pat- ity (GR) [1]. The curvature also has a close connection to tern in redshift and angular directions, respectively [23–26]. the physics of the early Universe. An inflationary universe The BAO experiments are shown to be robust against various scenario predicts that the “apparent” curvature, which we can astrophysical systematic effects such as the galaxy bias un- infer from an observable universe, should appear to be close certainty [27, 28]. The current state-of-the-art measurements to a flat geometry (Ω ' 0), even if the exact value is nonzero K were done using the Sloan Digital Sky Survey III Baryon Os- [2, 3]. If the Universe arose from the decay of a false vacuum cillation Spectroscopic Survey [29–31], achieving a percent via quantum tunneling, it leads to an open geometry (K < 0 precision of the distance measurement at z ' 0:57. Various or Ω > 0) [4–8]. In particular, if the Universe began with K wide-area spectroscopic galaxy surveys are planned that aim “large-field inflation” [9] – which predicts that the primordial to achieve precise BAO measurements up to higher redshifts: gravitational wave is as large as can be observed in the cos- the Subaru Prime Focus Spectrograph (PFS) [32], the Dark mic microwave background (CMB) anisotropies – or if the Energy Spectrograph Instrument (DESI) [33], the ESA Eu- universe began with “just enough inflation” in a landscape or clid satellite mission [57], and the NASA WFIRST mission multiverse picture, the curvature can be large enough to be −4 −2 [34]. measurable (ΩK ∼ 10 –10 ) [10–14]. Further, an addition of the super-curvature perturbation in an open-inflation sce- The BAO method is unique in that it can constrain the radial nario might resolve the large-scale CMB anomalies [15–17]. (more exactly the Hubble expansion rate) distance as well as On the other hand, if a closed curvature (K > 0 or ΩK < 0) is the angular diameter distance at the redshift of the galaxy sur- found, it gives rise to a challenge for the inflationary scenario: vey, while other geometrical probes such as supernovae and arXiv:1508.02469v3 [astro-ph.CO] 15 Mar 2018 the Universe needs to emerge from a specific initial condition gravitational lensing can probe the luminosity or angular dis- of closed curvature [7, 18]. Thus it is important to constrain tances (and not the radial distance). The relation between the the curvature from cosmological observations in order to ob- radial and angular diameter distances is purely geometrical tain a clue to the physics of the early universe [also see Ref. and specified by the curvature; if the Universe has a nonzero 19, for a thorough review of various cosmological probes]. curvature, the two distances differ. The relation holds for any Observations of the CMB have led to the precise measure- theory of gravity or dark energy, and rests on the metric theory ment of the angular diameter distance to the last scattering sur- of a homogeneous and isotropic space, which is a maximally face, preferring a flat geometry as expected in an inflationary symmetric spacetime described by the Friedmann-Robertson- scenario [e.g., 20, 21]. However, the constraint rests on the use Walker (FRW) metric [1]. Hence the purpose of this paper is of the standard cosmological model such as the cosmological to estimate the fundamental accuracy of estimating the curva- constant, cold-dark-matter dominated model (ΛCDM model), ture parameter with the BAO experiments [see Refs. 35, 36, based on GR and the nearly adiabatic initial conditions. If for the previous works based on the similar motivation]. To 2 do this, we will assume a cosmic-variance-limited galaxy sur- the existence of the BAO scale in the distribution of galaxies) vey, namely full-sky coverage and a sufficiently high number or any model of dark energy. of density of sampled galaxies. The structure of this paper is as follows. In Sec. II we dis- cuss the methodology based on the Fisher matrix information B. Estimator of the curvature and its covariance formalism, and briefly review the BAO method. In Sec. III we show the main results. Section IV is devoted to discus- Suppose we have the BAO distance measurements in Ns sion. Unless stated otherwise, we will adopt as fiducial model redshift bins, without any gap, over a range of redshift from a flat ΛCDM cosmology with Ωm0 = 0:27, ΩΛ = 0:73 and the today up to a maximum redshift zmax: Dˆ A(zi) and Dˆ H(zi) for Hubble parameter h = 0:71. i = 1; 2;:::; Ns (z1 = 0 and zNs = zmax). Here DH(z) is the comoving Hubble distance: DH(z) ≡ 1=H(z). The radial distance at the ith redshift bin zi can be estimated II. GEOMETRICAL ESTIMATION OF THE CURVATURE by combining the measured Hubble distances over a redshift WITH BAO DISTANCES range z = [0; zi] (see Eq. 1): X A. Cosmological distances Dˆ C;i ' Dˆ H; j∆z j; (4) z j<zi We assume that the Universe is statistically isotopic and where we have introduced the notations, Dˆ C;i ≡ Dˆ C(zi) and homogeneous. The spacetime structure of such an universe Dˆ H;i ≡ Dˆ H(zi). In practice one might want to use a more so- is described by the FRW metric [1]. With the metric theory, phisticated method to estimate Dˆ C(z) to avoid inaccuracy due solving the light propagation in an expanding universe yields to the discrete summation, e.g. by assuming that H(z) is mod- a relation between cosmological distances and redshift. The eled by a polynomial function of redshift and then estimating comoving radial distance is given in terms of the integral of the coefficients from fitting of the function to the measured the Hubble expansion rate: H(z). Here we assume a discrete summation for simplicity. Z z dz0 From Eq. (3), an estimator of the curvature parameter from D (z) ≡ ; (1) measurements of the radial and angular diameter distances at C H(z0) 0 the ith redshift bin can be given as where the Hubble expansion rate is given by the time deriva- Dˆ − Dˆ tive of the scale factor as H ≡ dln a=dt, and z is the redshift, Kˆ ≡ 6 A;i C;i : (5) i ˆ 3 given as 1 + z ≡ 1=a where we employed the convention DC;i a(t0) = 1 today. The metric theory also gives a geometrical Combining the measurements at different redshift bins, we relation between the radial and angular diameter distances: can define the chi-square (χ2) to estimate the curvature pa- 8 p rameter as > 1 > p sinh −KDC(z)(K < 0); > Ns > −K X h i−1 <> 2 ˆ KK ˆ DA(z) = DC(z)(K = 0); (2) χ ≡ (Ki − K) C (K j − K); (6) > p i j > 1 i; j=1 > p sin KDC(z)(K > 0) : K where K is the underlying true curvature, treated as a free 2 CKK −1 The different equations are for different geometries of the Uni- model parameter in the χ fitting. [ ] denotes the in- CKK verse; open, flat and close geometries, respectively. The cur- verse of the covariance matrix defined as 2 3 vature K is in units of 1=distance2. The CMB experiments CKK D i j 1 DA DA 1 6 A; j 1 7 DA DC = C − 63 − 2 7 C [e.g., Ref.
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