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Weighing the Universe with the Cosmic Microwave
Background

###### Abstract

Variations in , the total density of the Universe, leave a clear and distinctive imprint on the power spectrum of temperature fluctuations in the cosmic microwave background (CMB). This signature is virtually independent of other cosmological parameters or details of particular cosmological models. We evaluate the precision with which can be determined by a CMB map as a function of sky coverage, pixel noise, and beam size. For example, assuming only that the primordial density perturbations were adiabatic and with no prior information on the values of any other cosmological parameters, a full-sky CMB map at angular resolution and a noise level of per pixel can determine with a variance of 5%. If all other cosmological parameters are fixed, can be measured to better than 1%.

###### pacs:

98.70.V, 98.80.C^{†}

^{†}preprint: CU-TP-703 SU-4240-614 POP-629 CfA-4136 astro-ph/9507080

Determination of the geometry of the Universe remains perhaps the most compelling problem in cosmology. Alternatively stated, what is the mean total energy density of the Universe? The answer to this question will reveal the ultimate fate of the Universe. If the density (in units of the critical density , where is the Hubble constant) is greater than unity, the Universe is closed and will eventually recollapse; if it is less than unity, the Universe will expand forever; and if , the expansion will asymptotically decelerate to zero.

Theoretical considerations favor a critical () Universe, and inflation provides a generic mechanism for obtaining . However, luminous matter provides less than one percent of this mass. Various inferences of by dynamical means have hinted at substantial amounts of unseen mass, but most traditional methods of determining are plagued by systematic uncertainties. Furthermore, virtually all dynamical methods of obtaining give the mean density in only nonrelativistic matter, and thus cannot discriminate between an open Universe and a flat Universe that is dominated by vacuum energy (i.e., a cosmological constant).

Recently, it was proposed that temperature anisotropies in the cosmic microwave background (CMB) might be used to determine the geometry of the Universe [1]. Features (known as “Doppler peaks,” or more accurately as acoustic peaks) in the CMB angular power spectrum result from acoustic oscillations in the photon-baryon fluid before the photons decouple. The characteristic wavelength of these fluctuations is the sound horizon at decoupling (the distance an acoustic disturbance propagates from until decoupling), which subtends an angular scale on the sky today of . The dependence on arises directly from the geometry of the Universe, and this angular scale is largely independent of other cosmological parameters. Thus, the location of the first Doppler peak provides a robust determination of . A CMB map with fine angular resolution also constrains the other cosmological parameters by measuring the angular locations and amplitudes of the higher Doppler peaks.

In this paper, we evaluate the precision with which can be determined with high-resolution CMB maps [2]. We work within the context of models with adiabatic primordial density perturbations, although similar arguments apply to isocurvature models as well [3], and we expect the power spectrum to distinguish clearly the two classes of models. We also briefly consider what information on other cosmological parameters the CMB can provide.

A given cosmological theory makes a statistical prediction about the distribution of CMB temperature fluctuations, expressed by the angular power spectrum

(1) |

where is the fractional temperature fluctuation in the direction , are the Legendre polynomials, and the brackets represent an ensemble average over all observers and directions. The mean CMB temperature is [4]. Since we can only observe a single microwave sky, the observed multipole moments will be distributed about the mean value with a “cosmic variance” ; no measurement can determine the to better accuracy than this variance.

We consider an experiment which maps a fraction of the sky with a gaussian beam with full width at half maximum and a pixel noise , where is the detector sensitivity and is the time spent observing each pixel. We adopt the inverse weight per solid angle, , as a measure of noise that is pixel-size independent[5]. Current state-of-the-art detectors achieve sensitivities of , corresponding to an inverse weight of for a one-year experiment. Realistically, however, foregrounds and other systematic effects may increase the noise level; conservatively, will likely fall in the range . Treating the pixel noise as gaussian and ignoring any correlations between pixels, estimates of can be approximated as normal distributions with a variance (modified from Ref. [5])

(2) |

Given a spectrum of primordial density perturbations, the are obtained by solving the coupled equations for the evolution of perturbations to the spacetime metric and perturbations to the phase-space densities of all particle species in the Universe. We consider models with initial adiabatic density perturbations filled with photons, neutrinos, baryons, and collisionless dark matter; this includes all inflation-based models. We begin with approximate analytic solutions for the scalar [6] and tensor [7] metric perturbations. Our calculation includes polarization [8], scale dependence of the initial perturbation spectrum [9], and the large-angle integrated Sachs-Wolfe effect from a cosmological constant [10]. To a good approximation, reionization can be parameterized by the optical depth to the surface of last scatter [1]; anisotropies on scales much smaller than the horizon at reionization are suppressed by while those on larger scales are unaffected. The geometry of the Universe is then accounted for by shifting the moments, [1], and approximating the large-angle integrated Sachs-Wolfe effect in an open universe [11]. We do not here account for massive neutrinos (hot– or mixed– dark-matter models), but the power spectrum is altered only slightly by trading some of the nonrelativistic matter for neutrinos and our results should be unchanged[12].

The CMB power spectrum depends upon many parameters. In the present analysis we include the following set: the total density ; the Hubble constant, ; the cosmological constant in units of the critical density, ; the power-law indices of the initial scalar- and tensor-perturbation spectra, and ; the amplitudes of the scalar and tensor spectra, parameterized by , the total CMB quadrupole moment, and , the ratio of the tensor and scalar quadrupole moments; the optical depth to the surface of last scatter, ; the deviation from scale invariance of the scalar perturbations, ; and the effective number of light-neutrino species at decoupling, . Thus for any given set of cosmological parameters , we can calculate the mean multipole moments . ; the density of baryons in units of the critical density,

We now wish to determine the capability of CMB maps to determine these cosmological parameters. The answer to this question will depend on the measurement errors , and on the underlying cosmological theory. If the actual parameters describing the Universe are , then the probability distribution for observing a CMB power spectrum which is best fit by the parameters is

(3) |

where the curvature matrix is given approximately by

(4) |

with as given in Eq. (2). The covariance matrix is an estimate of the standard errors that would be obtained from a maximum-likelihood fit to data: the variance in measuring the parameter (obtained by integrating over all the other parameters) is approximately . If some of the parameters are known, then the covariance matrix for the others is determined by inverting the submatrix of the undetermined parameters. For example, if all parameters are fixed except for , the variance in is simply . In previous work, variances were estimated for small subsets of the parameters with Monte Carlo calculations [5, 13]; the present approach can be used to reproduce these results.

Fig. 1 displays the variance in as a function of the beam width for different noise levels and for . For different values of , replace and scale by [c.f., Eq. (2)]. The underlying model assumed here for the purpose of illustration is “standard CDM,” given by , where K is the COBE normalization [14]. The solid curves show the obtained by inversion of the full curvature matrix for , , and . These are the sensitivities that can be attained at the given noise levels with the assumption of uniform priors (that is, including no information about any parameter values from other observations). The dotted curves show the obtained by inversion of the - submatrix of ; this is the variance in that could be obtained if all other parameters except the normalization were fixed, either from other observations or by assumption. Realistically, the precision obtained will fall somewhere between these two sets of curves. Other underlying models, including low- models, give similar sensitivities. Although parameters other than will have some weak effect on the position of the first Doppler peak, they will also alter the power spectrum at smaller angular scales. Therefore, the higher multipole moments accessible with smaller beam widths help constrain the other parameters and make the determination of from the location of the first Doppler peak more precise.

Early reionization tends to wash out the structure of the power-spectrum features, decreasing the precision of the parameter estimates. To illustrate this effect, the curves in Fig. 2 show the same results as in Fig. 1, but for a reionized model in which . As expected, the sensitivity to decreases, although it remains significant even for as large as one half.

At , nonlinear effects become significant and linear power-spectrum calculations become unreliable. Therefore, we extend the sum in Eq. (4) only up to . With improved calculations, the sensitivities at small beam widths could conceivably be improved.

We have also investigated the sensitivity of CMB mapping experiments to the other cosmological parameters listed above. Our results suggest that a map with angular resolution may also provide interesting constraints to with minimal assumptions, and to the other parameters with reasonable priors. In particular, the experiments should be able to distinguish between a flat matter-dominated Universe and a flat cosmological-constant–dominated Universe. These results will be presented in detail elsewhere [15].

Figs. 1 and 2 estimate the probability of observing a set of parameters given an underlying model. Actual data will require solution of the inverse problem, estimating the probability of an underlying model given the data. We are currently exploring the accuracy with which all of the above parameters can be determined given a simulated data set [15, 16] and to what extent parameter degeneracy in current experiments can be resolved [17]. Preliminary results show that the true maximum of the likelihood function can be recovered with good accuracy from a parameter search routine, and that the errors in approach the precision obtained here.

The numerical results presented here demonstrate that can be determined by realistic next-generation satellite experiments with a precision on the order of a few percent. Such a measurement will greatly solidify our knowledge of the gross properties of the Universe, have crucial bearing on the dark-matter and age problems, and will provide a stringent test of the inflationary hypothesis.

We would like to thank Scott Dodelson, Lloyd Knox, Michael Turner, and members of the MAP collaboration for helpful discussions. This work is supported in part by the D.O.E. under contracts DEFG02-92-ER 40699 at Columbia University and DEFG02-85-ER 40231 at Syracuse University, by the Harvard Society of Fellows, by the NSF under contract ASC 93-18185 (GC3 collaboration) at Princeton University, and by NASA under contract NAGW-2448 and under the MAP Mission Concept Study Proposal at Princeton University. M.K. acknowledges the hospitality of the NASA/Fermilab Astrophysics Center where part of this work was completed.

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