produced by scalar perturbations and the latter by tensor perturbations, such as those due to gravitational waves in the primordial universe. So if we are able to ...
Theoretical Foundations of Primordial Gravitational Waves Printed in the Cosmic Microwave Background and its Observational Status. Alexander Bonilla Rivera1.
1
Departamento de Física y Astronomía, Facultad de Ciencias, Universidad de Valparaíso, Valparaíso, Chile. ABSTRAC
The anisotropy study cosmic microwave background (CMB) is one of the main observational tools for modern cosmology. However, alongside the study of the thermal fluctuations of the CMB are other equally important information, which is known as the polarization of the CMB. The inflationary model predicts that the CMB is linearly polarized and the physical mechanism of this polarization is studied from the Thompson scattering, the dominant process on the surface of last scattering. There are basically two types of polarization called E and B modes, the first produced by scalar perturbations and the latter by tensor perturbations, such as those due to gravitational waves in the primordial universe. So if we are able to measure these types of polarization will have an entry to the study of the inflationary epoch. This paper presents the main physical mechanisms that support theoretically the polarization of the CMB due to primordial gravitational waves (PGW) and the revision of the main observables, grouped in so-called Stokes parameters (Q, U, I), Which brings us information to achieve the contrast the angular power spectrum produced by the polarization of the CMB, which shows to be in excellent agreement with the model 𝚲𝑪𝑫𝑴.
THE CMB ANISOTROPY POWER SPECTRUM The CMB consist of photons that last interacted with the matter (z~1100) 5. Since the universe must have already been inhomogeneous at this time, in order for the structures present in the universe today to be able to form, it is expect that these spatial inhomogeneous are reflect in small anisotropies of the CMB (Fig1). As CMB anisotropies are small, they can be treated nearly completely within linear cosmological perturbation theory3. Since the CMB anisotropies are a function on a sphere, they can be expanded in spherical harmonics: ∆𝑇 𝑇
𝑚 𝑙=1
=
𝑚 =𝑙 𝑚 =−𝑙
𝑎𝑙𝑚 𝑌𝑙𝑚 𝑛 ,
1.
where 𝑛0 corresponding to unpolarized isotropic thermal radiation, with 𝑛0 = 𝑒𝑥𝑝 ℎ𝜈 𝐾𝑇 − 1 −1 , which depends only on the photon frequency and corresponds to the Plank spectrum. The Boltzmann equation of radiative transfer written in terms of 𝑛 𝜂, 𝑥 𝛼 , 𝜃, 𝜑 is: 𝜕𝑛 𝜕𝜂
𝜕𝑛
𝜕 𝑛 𝜕𝜈
+ 𝑒 𝛼 𝜕 𝑥 𝛼 = 𝜕𝜈
𝜕𝜂
1
−𝑞 𝑛−𝑗 ,
𝑗 = 4𝜋
𝜋 2𝜋 0 0
𝑛𝑃 𝑆𝑖𝑛𝜃𝑑𝜃𝑑𝜑
6.
where 𝑞 = 𝜎𝑇 𝑁𝑒 𝑎, 𝑎 is the cosmological scale factor, 𝑗 is the scattering term and 𝑁𝑒 is the commoving number density of free electrons and 𝜎𝑇 Thomson cross section.
where Δ𝑇 = 𝑇 − 𝑇0 and 𝑇0 is the mean temperature on the sky. We define the power spectrum through the relation 𝑎𝑙𝑚 . 𝑎𝑙∗′𝑚 ′ = 𝛿𝑙𝑙 ′ 𝛿𝑚 𝑚 ′ 𝐶𝑙 ,
𝐶𝑙 = 𝐶𝑙𝑚
2
1
= 2𝑙+1
𝑚 =𝑙 𝑚 =−𝑙
𝑎𝑙𝑚 2 ,
2.
where the 𝐶𝑙 are the CMB power spectrum (Fig 2).
Fig 3. The scalar (left) and tensor (right)
Fig4. Measurements of the curl-mode polarization
The coupling of the gravitational waves to the radiation is manifested in the first term of the right side of Eq 6, like2 1 𝜕𝜈 𝜈 𝜕𝜂
Fig 1. CMB Polarization map6.
Fig 2. CMB Power Spectrum6.
PRIMORDIAL GRAVITATIONAL WAVES In a flat universe filled with dustlike matter, the linearized Einstein equations will have the following solution for the tensor metric perturbations describing gravitational waves1 𝛽 ℎ𝛼
=
1 𝜕 𝜂 𝜕𝜂
1
𝛽 𝐷 𝜂 𝛼
𝛽 𝐷𝛼
,
= 2𝜋
−1
𝛽 𝑓𝛼 𝑒 −𝑖 𝑘𝑥 −𝜔𝜂
3
𝑑 𝑘,
3.
𝛽
where is the conformal time and the tensor 𝐷𝛼 constitutes a superposition of plane waves. At an earlier era 𝑝 = 𝜀 3 the solution for the gravitational waves 𝛽 𝛽 𝛽 𝛽 taken the form ℎ𝛼 = ℎ𝑃𝛼 𝑒 −𝑖 𝑘𝑥 −𝑘𝜂 , 𝑓𝛼 = 𝑓𝑃𝛼 , 𝛽 where 𝑃𝛼 denotes the gravitational wave polarization tensor. POLARIZATION There exist two types of polarization signals: the so called E-type polarization which has positive parity, and B-type polarization which is parity odd (Fig3). Scalar perturbations only produce E-type polarization, while tensor perturbations, gravity waves, produce both, E- and B-type. A typical CMB anisotropy and polarization spectrum as it is expected from inflationary models is shown in Fig. The polarization tensor is defined as 𝑃𝑖𝑗 = 𝑃𝑎𝑏 𝜖𝑖𝑎 𝜖𝑗𝑏 ,
𝑃𝑎𝑏 =
1 2
𝛼 𝐼 + 𝑈 + 𝑉 + 𝑄 𝜎𝑎𝑏 ,
4.
𝛼 where 𝑃𝑎𝑏 = 𝐸𝑎∗ 𝐸𝑏 and 𝜎𝑎𝑏 are the Pauli matrices and I , U, V and Q are the 4 Stokes parameters .
TRANSFER EQUATION We now turn to the imprint of tensor gravitational waves on CMB polarization using the equation of radiative transport. By definition2: 𝑐2
𝑛 = ℎ𝜈 3 𝐼 = 𝑛0 + 𝑛0 𝛿𝑛,
𝐼 = 𝐼𝑥 , 𝐼𝑦 , 𝑈 ,
𝑛0 = 𝑛0 1,1,0 ,
5.
1 𝜕 ℎ 𝛼𝛽
=2
𝜕𝜂
𝑒𝛼 𝑒𝛽 ,
𝑛 = 𝑛0 + 𝑒𝑥𝑝 −𝑖𝑘𝑥 + 𝑖𝑘𝜂 𝑛1
7.
The effect of the GW upon the propagation of radiation amounts to shifting the frequencies of photons along the line of sight, and is described by the Eq 6. The perturbed solution of de Eq 6, linearized with respect to ℎ𝛼𝛽 , will be Eq 7. where 𝑛1 𝜂, 𝜈 is a vector that depend of 𝜂 and 𝜈. OBSERVATIONAL STATUS If the fluctuations in CMB intensity are printed by scalar perturbations, one would only expect primordial E-modes in the CMB polarization. However, vector and tensor perturbations, like those due to gravitational waves in the primordial Universe, are mechanisms that could generate primordial B-modes. In particular, the energy scale 𝑉 at which inflation occurred can be expressed in terms of 𝑟, the ratio of tensor to scalar contributions to the power spectrum, as Δ2 k 0
𝑟 = Δ2h
R
k0
= 0.001 𝑉 1016 𝐺𝑒𝑉
4
8.
Where Δ2R k 0 and Δ2h k 0 are the amplitudes of the scalar and primordial power spectrum. The power spectrum for the WMAP best fit cosmological model with 𝜏 = 0.17 and 𝑟 = 0.1 (Fig4, solid line). CONCLUSIONS Because of the importance of detecting primordial gravitational waves there is a huge interest to develop ground-based experiments to measure (or constrain) the amplitude of B-modes power spectrum of the CMB polarization. Currently, DASI, BOOMERANG, CBI, and CAPMAP have detected E-mode polarization2. However there are no current detections of the E-mode polarization for 𝑙 < 100 where the signature of gravitational waves will be manifest. Biografía 1. 2. 3. 4. 5. 6.
G. Polnarev, Astron. Zh. 62, 1041-1052 (November-December 1985). B. G. Keating, A. G. Ponarev, N. J. Miller, D. Baskaran. arxiv: astro-ph/0607208v1, 10 jul 2010. E. W. Kolb, M. S. Turner. “The Early Universe”. Addison- Wesley Publishing Company, 1989. J. D. Jackson.“Classical Electrodynamics“. Wiley & Sons, 1999, 3rd ed. Ruth Durrer, “The Cosmic Microwave Background”. Cambridge University Press, 2008. http://lambda.gsfc.nasa.gov/product/map/current/
