Difference between revisions of "CMB anisotropy"
From Universe in Problems
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=== Problem 1 === | === Problem 1 === | ||
− | <p style= "color: #999;font-size: 11px">problem id: </p> | + | <p style= "color: #999;font-size: 11px">problem id: per25</p> |
− | + | Obtain the equation of motion for photon in metrics ($ds^2=a^2(\eta)[(1+2\Phi)d\eta^2-(1-2\Phi)\delta_{ij}dx^idx^j]$) in linear approximation in $\Phi$. | |
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− | <p style="text-align: left;"></p> | + | <p style="text-align: left;">Equation of geodesics describing the propagation of radiation in space with arbitrary curvature can be represented as: |
+ | $$ | ||
+ | \frac{dx^\alpha } | ||
+ | {d\lambda } = p^\alpha,\; \frac{dp_\alpha } | ||
+ | {d\lambda} = \frac{1} | ||
+ | {2}\frac{\partial g_{\gamma \delta }} | ||
+ | {\partial x^\alpha }p^\gamma p^\delta, | ||
+ | $$ | ||
+ | where $\lambda$ is an arbitrary affine parameter, which is taken along geodesic. Since photon has zero mass, the first integral of this equation is: | ||
+ | $$ | ||
+ | p^\alpha p_\alpha = g^{\mu \nu }p_\mu p_\nu = 0. | ||
+ | $$ | ||
+ | Using this relation, zero component can eliminated from the equations of geodesics: | ||
+ | $$ | ||
+ | p^0 = \frac{1} | ||
+ | {a^2}\left( \sum\limits_{i = 1}^3 p_i^2 \right)^{1/2} = \frac{p}{a^2} | ||
+ | $$ | ||
+ | and | ||
+ | $$ | ||
+ | p_0 = \left( 1 + 2\Phi\right)p. | ||
+ | $$ | ||
+ | Hence, | ||
+ | $$ | ||
+ | \frac{dx^i} | ||
+ | {d\eta } = \frac{p^i} | ||
+ | {p^0} = \frac{ - \frac{1} | ||
+ | {a^2}\left( 1 + 2\Phi \right){p_i}} | ||
+ | {p^0} = r^i\left( 1 + 2\Phi\right), | ||
+ | $$ | ||
+ | where $r^i = - \frac{p_i}{p^0}$. Expressing $p^i$ and $p^0$ through $p$ and substituting metrics into the second equation while keeping only linear terms in $\Phi$ one obtains: | ||
+ | $$\frac{dp_\alpha } | ||
+ | {d\eta} = \frac{1} | ||
+ | {2}\frac{\partial g_{\gamma \delta }} | ||
+ | {\partial x^\alpha }\frac{p^\gamma p^\delta } | ||
+ | {p^0} = 2p\frac{\partial \Phi } | ||
+ | {\partial x^\alpha }. | ||
+ | $$</p> | ||
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− | <div id=""></div> | + | <div id="per26"></div> |
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=== Problem 1 === | === Problem 1 === | ||
− | <p style= "color: #999;font-size: 11px">problem id: </p> | + | <p style= "color: #999;font-size: 11px">problem id: per26</p> |
− | + | For the Universe, dominated by a substance with equation of state $ p = w \rho $, connect in the first approximation the fluctuations of the gravitational potential of the CMB with $ \Phi$. | |
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− | <p style="text-align: left;"></p> | + | <p style="text-align: left;">Using the invariant $aT = const$: |
+ | $$ | ||
+ | \frac{\delta T}{T} =-\frac{\delta a}{a}. | ||
+ | $$ | ||
+ | In case of $p = w\rho ,\; a\left( t \right) = t^{\frac{2} | ||
+ | {3\left( 1 + w \right)}}$ and, hence, | ||
+ | $$ | ||
+ | \frac{\delta a} | ||
+ | {a} = \frac{2} | ||
+ | {3\left(1 + w\right)}\frac{\delta t}{t}. | ||
+ | $$ | ||
+ | In the case of a weak gravitational field (in this approximation, we assume that the density perturbations generate a small perturbation of the metric, which is valid for the time at which we see these disturbances), the true time $\tau$ is connected to coordinate time $t$ as | ||
+ | $$ | ||
+ | d\tau=\sqrt{g_{00}}dt\approx\sqrt{1 + 2\Phi } dt \approx \left(1+\Phi\right)dt, | ||
+ | $$ | ||
+ | where $\Phi$ is a Newtonian gravitational potential and, thus, ${\delta t}/t \simeq \Phi$: | ||
+ | $$ | ||
+ | \left(\frac{\delta T}{T} \right)_e = - \frac{2}{3\left( 1 + w\right)}{\Phi_e} | ||
+ | $$ | ||
+ | where subscript $e$ denotes the moment of emission. | ||
+ | When light is propagating in expanding Universe, the relation $\omega (t) \propto a(t)^{-1}$ holds. Using the same arguments in Newtonian approximation, one could write | ||
+ | $$ | ||
+ | \left( \frac{\delta T} | ||
+ | {T} \right)_0 = \left(\frac{\delta T} | ||
+ | {T}\right)_e + \Phi _e | ||
+ | $$ | ||
+ | where subscript $0$ denotesthe moment of detection. Finally, | ||
+ | $$ | ||
+ | \left(\frac{\delta T} | ||
+ | {T} \right)_0 = \left(\frac{1 + 3w} | ||
+ | {3 + 3w} \right)\Phi _e | ||
+ | $$</p> | ||
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=== Problem 1 === | === Problem 1 === | ||
<p style= "color: #999;font-size: 11px">problem id: </p> | <p style= "color: #999;font-size: 11px">problem id: </p> | ||
− | + | Estimate the spatial scale of Silk effect. Assume that at the temperatures we are interested in, photon changes direction randomly, and its energy does not change when scattering on electrons. | |
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<div class="NavHead">solution</div> | <div class="NavHead">solution</div> | ||
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− | <p style="text-align: left;"></p> | + | <p style="text-align: left;">In this approximation, the oscillations of the photon component |
+ | smooth on scales smaller than the distance at which the photon diffuses | ||
+ | in characteristic period of evolution. If considering the before recombination era this time is the Hubble time. | ||
+ | |||
+ | In this ''random walk'' approximation Silk scale $\lambda_S$ can be estimated as geometric average of mean free path of a photon $\lambda_\gamma$ and horizon scale $l_H$ (or Hubble time $t_H\sim H^{-1}$). The number collisions between photon and electrons during the Hubble time is estimated as $t_H/\lambda_\gamma$ and distance between collisions is of order of $\lambda_\gamma$, so that a photon diffuses at distance $$\lambda_S \sim \sqrt{N}\lambda_\gamma =\sqrt{\frac{t_H}{\lambda_\gamma}}\lambda_\gamma = \sqrt{\lambda_\gamma l_H}$$ during the Hubble time. Hubble parameter at recombination era is | ||
+ | $$H(z_r)=H_0\sqrt{\Omega_{m0}(1+z_r)^3+\Omega_{r0}(1+z_r)^4}\simeq 5\, \mbox{Mpc}^{-1}.$$ | ||
+ | |||
+ | Densities of free electrons and protons coincide and the latter before the resombination is about 75 \% of barion density: other barions (and electrons) are contained in helium atoms, which formed in the Universe somewhat earlier. Thus, before the start of resombination the electrons density satisfies | ||
+ | $$n_e(z)=0.75 \frac{\rho_b(z)}{m_p}=6\cdot 10^{-4}\Omega_b (1+z)^3 \mbox{sm}^{-3}.$$ | ||
+ | At the beginning of recombination the number density of free electrons is | ||
+ | $n_e(\eta_r)=230\,\mbox{sm}^{-3}.$ | ||
+ | Hence, | ||
+ | $$\lambda_S(\eta) \simeq \sqrt{\frac{1}{\sigma_{_T}}n_e(\eta) H(\eta)},$$ | ||
+ | which gives $\lambda_S(\eta_r) \simeq 0.02 \,\mbox{Mpc.}$</p> | ||
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=== Problem 1 === | === Problem 1 === | ||
<p style= "color: #999;font-size: 11px">problem id: </p> | <p style= "color: #999;font-size: 11px">problem id: </p> | ||
− | + | Estimate the angular scale of the CMB anisotropy due to the Silk effect. | |
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<div class="NavHead">solution</div> | <div class="NavHead">solution</div> | ||
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− | <p style="text-align: left;"></p> | + | <p style="text-align: left;">Physical distance $R(z)$ is connected to the angular diameter distance $R_a(z)$ as |
+ | $$R_a(z)=\frac{1}{1+z}R(z),$$ where $$R(z)=R_H\int_0^{z}\frac{dz}{\sqrt{\Omega_{m0}(1+z)^3+\Omega_{\Lambda 0}}},$$ and $R_H$ is Hubble radius. | ||
+ | Using the relation $\theta \simeq \frac{\lambda_S}{R_a(z_r)}$ one can obtain $\theta \simeq 10^{-3}$.</p> | ||
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