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Baryon Acoustic Oscillations. Equation and Physical Interpretation Scientia et Technica Año XXIII, Vol. 23, No. 02, junio de 2018. Universidad Tecnológica de Pereira. ISSN 0122-1701 263 Baryon Acoustic Oscillations. Equation and physical interpretation Baryon Acoustic Oscillations. Equation and physical interpretation. Leandro Manuel Pardo Calderón Observatorio Astronómico Nacional, Universidad Nacional, Bogotá D.C., Colombia [email protected] Abstract —Baryon Acoustic Oscillations are a a harmonic oscillator equation. This fact, together phenomenon occurred before matter-radiation with some approximate solutions will make easier to decoupling, characterized because the baryonic matter understand its physical interpretation. Software perturbation presents oscillations, as the name suggests. CAMB was used to complement the investigation of These perturbations propagate like a pressure wave on baryon acoustic oscillations phenomenon through the photon-baryon fluid produced by gravitational potentials, which join the baryonic matter, and analysis of simulations and graphics. collisions of baryons and photons, which scatter it. This paper shows the Baryon Acoustic Oscillations equation II. COSMOLOGICAL and it provides its physical meaning. Besides, it PERTURBATIONS presents software CAMB as a tool to find BAO equation solutions and support for its physical The universe must contain inhomogeneities that description. explain grouping of matter, like galaxies clusters (figure 1). These inhomogeneities occurs on an Key Word —Acoustic oscillations, baryons, cosmological idealized universe, thus, it is convenient first to perturbations, Newtonian gauge. discuss about the ideal universe. I. INTRODUCTION First models about universe appeared few years after Albert Einstein published his General Relativity Theory. They described an isotropic and homogeneous expanding universe. However, since the universe contains inhomogeneities, which are the cause of great structures formation, it was necessary to develop a Cosmological Perturbation Theory. To be able to obtain a refinement when studying the structures formation, it is necessary to include Figure 1. Inhomogeneities in the universe. Picture taken baryonic matter. Baryon Acoustic Oscillations from page https://www.universetoday.com/54756/what-is- phenomenon is an approach to investigate baryonic the-big-bang-theory/. matter behavior. A. Ideal universe This paper will show the baryon acoustic oscillations equation and its physical interpretation, after a short The ideal universe is depicted by a couple of discuss about cosmological perturbation theory to equations known together as Friedmann Equations introduce the topic. Physical conditions of the [1]: universe that make baryon acoustic oscillations to exist are also explained. , (1) = + For this purpose, it will be showed how baryon acoustic oscillations equation can be approximated to Fecha de Recepción: 29 de noviembre de 2017 Fecha de Aceptación: 23 de junio de 2018 264 Scientia et Technica Año XXIII, Vol. 23, No. 02, junio de 2018. Universidad Tecnológica de Pereira. , (2) = − + 3 + C. Temperature perturbations where a is the scale factor , which only depends on The perturbations on the temperature can be cosmological time , , is the cosmological constant, observed now in the CMB (Cosmic Microwave and are the average density and pressure Background) and they have an order of . Since * respectively. Symbol prime represents derivatives the average temperature of CMB is , it10 means that with respect to cosmological time. The Friedmann temperature perturbations are of the 3 +order of some equations describe the evolution of a homogeneous . The temperature pertubations in CMB were and isotropic universe, which is in an increasing studied,+ by the first time at 1995 by COBE (Cosmic expansion. Background Explorer) satellite [5], but its observations has been improved by further project, From the terms involved in Friedmann equations it is being PLANCK the last one, in 2013 [6]. possible to define some useful parameters like Hubble rate , which is a measurement of the D. Perturbations on metric = universe expansion speed, Hubble radius is = Perturbations in space-time are described through the the distance for which universe expansion speed is metric: higher than light speed, and Hubble time , is the lapse needed by light to travel a Hubble =radius. −1 + 12 0 . (5) -./ = 0 7 B. Definition of cosmological perturbations 0 3 1 + 14"56 This metric is known as Newtonian gauge [7]. It As it was told before, the real universe must contain includes only scalar perturbations where and are inhomogeneities that explain grouping of matter. 2 4 These inhomogeneities are caused by perturbations in scalar functions which depend on space-time coordinates, with the conditions , . the universe. Because of, it is important to define the 828 9 1 848 9 1 cosmological perturbations. The newtonian gauge metric allows to describe the formation of great structures, where and play 2 4 We count with two different space-times. One of the function of perturbed gravitational potencials. them describes the ideal homogeneous and isotropic universe, ruled by Friedmann equations. The other III. BARYON ACOUSTIC one describes the real universe, which contains OSCILATIONS grouped matter. We can associate a point of real The baryon acoustic oscillations (BAO from now on) space-time with a point of ideal space-time via a is a phenomenon ocurred at the early times of mapping . This association is called a universe, before the decoupling of matter and gauge choice =. The ! mapping should be one to one, but radiation, where the perturbation of baryonic matter it is not unique and making a different mapping to propagated as a wave. This section shows the imply to do a gauge transformation. In this way we equation which rules BAO and the physical meaning can study the behavior of the real universe through of it. However, it is convenient, in a first place, to the information of ideal universe, which is already indicate the phsysical conditions of the universe at known enough. Now we can define the perturbations age of interest, which made posible the presence of [2, 3, 4] as the difference of the quantities of real BAO. universe and ideal universe, like, for example, perturbations on density and temperature: A. Physical conditions of the universe before matter-radiation decoupling . (3) " = −,$ "% =%−% Time of decoupling is placed at years after Also, we can define the fractional perturbations as Big-Bang . Before that, the universe300.000 temperature was following: around making the photons to be very energetic.3000 Photons + interacted with baryons, which, in & &' . (4) the cosmology context, refer not only protons but " = ,"' = $ ' also electrons, via Compton scattering . This strong tight coupled limit interaction is known as and it caused that photons and baryons moved together like an unique fluid, the photon-baryon fluid . Because of, Scientia et Technica Año XXIII, Vol. 23, No. 02, junio de 2018. Universidad Tecnológica de Pereira. 265 the model to describe this behavior is known as fluid This is a much more familiar equation. It is a damped approximation. Energetic photons were able to forced harmonic oscillator equation, where the scatter baryons and they avoid that proton and forced term is given by gravitational potentials. It is electrons to join into neutral atoms [8]. not strange to obtain this kind of equation if we consider that BAO phenomenon is produced by a The previous described conditions finished when the competition between pressure forces, produced by Hubble rate becomes higher than scatter rate. Compton scattering, which separate the baryonic Photons are not energetic enough to scatter baryons, matter, and gravitational potentials, which group it occurring the decoupling of matter and radiation (view figure 2 for a descriptive picture). Protons and electrons to join in neutral atoms and photons follow free ways, being possible to be observed currently in the CMB. B. The equation of baryon acoustic oscillations The equation that rules the BAO phenomenon is: = "<; + "<= + > ?@ "< Figure 2. Pressure on baryons in a gravitational potential 1 + well. This picture has been modified from page = http://background.uchicago.edu/~whu/power/bao.html. = −> 2 − 3 4= − 34; , 1 + (6) It creates oscillations which propagate on the photon- where overdots represent derivatives with respect to baryon fluid (figure 3). conformal time , which is written in terms of cosmological timeA as: D CD . (7) A = BE D Besides, is the photon-baryon ratio: $F (8) = $ G and is the sound speed, defined via photon-baryon ratio? as:@ . (9) ?@ = HIJ Thus, (6) is a second order linear differential equation for baryon fractional perturbation . It is written in terms of wave number , since" <it has Figure 3. Baryonic matter oscillates on the photon-baryon fluid. > This picture has been taken from page applied Fourier transform to it, indicating that BAO http://zuserver2.star.ucl.ac.uk/~mts/webpage_dev/BAOs.html. are analysed in term of lenght scales instead of spatial coordinates. The kind of movement for baryonic matter, described above, is analogue to the propagation of a gas C. Physical meaning of bao equation pressure wave, just like sound waves on the air. It justifies the term acoustic for BAO phenomenon, and To show the physical meaning of BAO equation (6) others derived, like sound speed, etc. we can rewritte it in a schematic way as: . (10) "<; + K"<= + L "< = M 266 Scientia et Technica Año XXIII, Vol. 23, No.
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