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Cosmic Microwave Background Theory Proc. Natl. Acad. Sci. USA Vol. 95, pp. 35–41, January 1998 Colloquium Paper This paper was presented at a colloquium entitled ‘‘The Age of the Universe, Dark Matter, and Structure Formation,’’ organized by David N. Schramm, held March 21–23, 1997, sponsored by the National Academy of Sciences at the Beckman Center in Irvine, CA. Cosmic microwave background theory J. RICHARD BOND* Canadian Institute for Advanced Research Cosmology Program, Canadian Institute for Theoretical Astrophysics, 60 Saint George Street, Toronto, ON M5S 3H8, Canada ABSTRACT A long-standing goal of theorists has been to ponent in gravitational wave tensor perturbations; the power 3 3 3 constrain cosmological parameters that define the structure spectra for these modes, F(k), is(k), GW(k) as a function formation theory from cosmic microwave background (CMB) of comoving wavenumber k. Sample initial and evolved power 2 anisotropy experiments and large-scale structure (LSS) ob- spectra for the gravitational potential 3F(k)([ dsFydln k, servations. The status and future promise of this enterprise is the rms power in each dln k band) are shown in Fig. 1. As the described. Current band-powers in ø-space are consistent with Universe evolves the initial shape of 3F is modified by a DT flat in frequency and broadly follow inflation-based characteristic scales imprinted on it that reflect the values of expectations. That the levels are ;(1025)2 provides strong cosmological parameters such as the energy densities of bary- support for the gravitational instability theory, while the Far ons, cold and hot dark matter, in the vacuum (cosmological Infrared Absolute Spectrophotometer (FIRAS) constraints on constant), and in curvature. Many observables can be ex- energy injection rule out cosmic explosions as a dominant pressed as weighted integrals over k of the power specta and source of LSS. Band-powers at ø * 100 suggest that the thus can probe both density parameters and initial fluctuation universe could not have re-ionized too early. To get the LSS of parameters. Cosmic Background Explorer (COBE)-normalized fluctua- The (linear) density power spectra, 3r(k) } k43F(k), are tions right provides encouraging support that the initial also shown in Fig. 1. In hierarchical structure formation fluctuation spectrum was not far off the scale invariant form models such as those considered here, the nonlinear wave- L *kNL 3 5 that inflation models prefer: e.g., for tilted cold dark matter number kNL(t), defined by 0 r(k)dln k 1, grows as the sequences of fixed 13-Gyr age (with the Hubble constant H0 universe expands. kNL(t) was in the galaxy band at redshift 3 5 6 . marginalized), ns 1.17 0.3 for Differential Microwave and is currently in the cluster band. At k kNL(t), nonlin- Radiometer (DMR) only; 1.15 6 0.08 for DMR plus the SK95 earities and complications associated with dissipative gas experiment; 1.00 6 0.04 for DMR plus all smaller angle processes can obscure the direct connection to the early experiments; 1.00 6 0.05 when LSS constraints are included universe physics. Most easily interpretable are observables L , as well. The CMB alone currently gives weak constraints on probing the linear regime now, k kNL(t0). CMB anisotropies V and moderate constraints on tot, but theoretical forecasts of arising from the linear regime are termed primary; as Fig. 1 future long duration balloon and satellite experiments are shows, these probe 3 decades in wavenumber. LSS observa- shown which predict percent-level accuracy among a large tions at low redshift probe a smaller, but overlapping, range. 1 ; fraction of the 10 parameters characterizing the cosmic We have hope that z 3 LSS observations, when kNL(t) was structure formation theory, at least if it is an inflation variant. larger, can extend the range, but gas dynamics can modify the relation between observable and power spectrum in complex ways. Secondary anisotropies of the CMB (see below), those THE THEORETICAL AGENDA associated with nonlinear phenomena, also probe smaller scales and the ‘‘gastrophysical’’ realm. Cosmic Microwave Background (CMB) as a Probe of Early Universe Physics Cosmic Parameters The source of fluctuations to input into the cosmic structure Even simple Gaussian inflation-generated fluctuations for formation problem is likely to be found in early universe structure formation have a large number of early universe physics. We want to measure the CMB [and large-scale parameters we would wish to determine (see next section): structure (LSS)] response to these initial fluctuations. The goal power spectrum amplitudes at some normalization wavenum- 3 3 3 is the lofty one of peering into the physical mechanism by ber kn for the modes present, [ F(kn), is(kn), GW(kn)]; n n n which the fluctuations were generated. The contenders for shape functions for the ‘‘tilts’’ [ s(k), is(k), t(k)], usually generation mechanism are (i) ‘‘zero point’’ quantum noise in chosen to be constant or with a logarithmic correction—e.g., n n y scalar and tensor fields that must be there in the early universe s(kn), d s(kn) dln k. [The scalar tilt for adiabatic fluctuations, if quantum mechanics is applicable and (ii) topological defects that may arise in the inevitable phase transitions expected in Abbreviations: CMB, cosmic microwave background; LSS, large-scale the early universe. structure; COBE, Cosmic Background Explorer; MAP, Microwave From CMB and LSS observations we hope to learn the Anisotropy Probe; DMR, Differential Microwave Radiometer; CDM, cold dark matter; LCDM, CDM with a cosmological constant; SCDM, following: the statistics of the fluctuations, whether Gaussian the standard CDM model; TCDM, tilted CDM; HCDM, CDM with or non-Gaussian; the mode, whether adiabatic or isocurvature hot dark matter; OCDM, open (negatively curved) CDM models; scalar perturbations, and whether there is a significant com- COMBA, Cosmic Background Radiation Archive; FIRAS, Far Infra- red Absolute Spectrophotometer; SZ, Sunyaev–Zeldovich; LDB, long- duration balloon; GW, gravity wave; HEMT, High Electron Mobility © 1998 by The National Academy of Sciences 0027-8424y98y9535-7$2.00y0 Transistor; Mpc, megaparsec. PNAS is available online at http:yywww.pnas.org. *e-mail: [email protected]. 35 Downloaded by guest on September 30, 2021 36 Colloquium Paper: Bond Proc. Natl. Acad. Sci. USA 95 (1998) FIG. 1. The bands in comoving wavenumber k probed by CMB primary and secondary anisotropy experiments, in particular by the satellites Cosmic Background Explorer (COBE), Microwave Anisot- ropy Probe (MAP), and Planck, and by LSS observations are con- trasted. Mpc, megaparsec (3.09 3 1022 m). The width of the CMB photon decoupling region and the sound crossing radius (Dtgdec, cstgdec) define the effective acoustic peak range. Sample (linear) 1y2 gravitational potential power spectra [actually 3F (k)] are also plot- ted. The region at low k gives the 4-yr Differential Microwave Radiometer (DMR) error bar on the F amplitude in the COBE regime. The solid data point in the cluster-band denotes the F constraint from the abundance of clusters, and the open data point at 21 10h Mpc denotes a F constraint from streaming velocities (for Vtot 5 1, VL 5 0). The open squares are estimates of the linear F power from current galaxy clustering data by ref. 1. A bias is ‘‘allowed’’ to (uniformly) raise the shapes to match the observations. The corre- 1y2 sponding linear density power spectra, 3r (k), are also shown rising to high k. Models are the ‘‘standard’’ ns 5 1 cold dark matter (CDM) model (labeled G50.5), a tilted (ns 5 0.6, G50.5) CDM (TCDM) model, and a model with the shape modified (G50.25) by changing the matter content of the Universe. n [ 3 y n 5 s(k) dln F dln k, is related to the usual index, ns,by s 2 ns 1.] The transport problem (see below) is dependent upon physical processes, and hence on physical parameters. A partial FIG. 2. The anisotropy data for experiments up to March 1997 (top list includes the Hubble parameter h, various mean energy panel) and optimal combined bandpower estimates (lower panels) are compared with secondary #, values (top panel) and various primary densities [V , V , VL, V , V ]h2, and parameters charac- tot B cdm hdm #, sequences. The kinematic Sunyaev–Zeldovich (SZ)#, is off scale terizing the ionization history of the Universe—e.g., the Comp- and the thermal SZ is low; compare the primary #, values. Dusty t ton optical depth C from a reheating redshift zreh to the present. emission from early galaxies may lead to high signals, but the power V , Instead of tot, we prefer to use the curvature energy parameter, is concentrated at higher and higher frequency. The next panels are V [ 2V k 1 tot, thus zero for the flat case. In this space, the Hubble sequences of 13-Gyr models with variations in the following param- y V 5 S V 2 1 2 eters: ns, 0.85 to 1.25 (panel 2 from the top); L, 0 to 0.87 (H0 from parameter h ( j( jh )) , and the age of the Universe, t0, are V V 2 50 to 90) (panel 3); k, 0 to 0.84 (H0 from 50 to 65) (panel 4); and functions of the jh . The density in nonrelativistic (clustering) V 2 # V 5V 1V 1V Bh , 0.003 to 0.05 (panel 5). Panel 6 shows that sample defect , particles is nr B cdm hdm. The density in relativistic V values from Pen, Seljak, and Turok (2) do not fare well compared with particles, er, includes photons, relativistic neutrinos, and decay- the current data; #, values from ref. 3 are similar. Panels 7 and 8 show V ing particle products, if any.
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