Physical Cosmology 18/5/2017

Transcription

1 Physical Cosmology 18/5/2017 Alessandro Melchiorri slides can be found here: oberon.roma1.infn.it/alessandro/cosmo2017

2 Summary If we consider perturbations in a pressureless matter component (Jeans length always zero) their growth depend on which kind of energy component is dominating the expansion. We have substantial growth only here!

3 Moving to Fourier Space In order to move to a more physical description let us consider an expansion in Fourier modes of the density contrast field: Each Fourier mode is given by:

4 Moving to Fourier Space Each Fourier component is a complex number, which can be written in the form: It is possible to show that assuming a linear perturbation theory, i.e. : And following a more formal approach, one gets, in the Newtonian regime, for a pressure less fluid:

5 Moving to Fourier Space This is the same equation we got for the density contrast of a sphere of radius R, but now it applies to each Fourier mode of a generic density contrast! The fact that we assumed linear perturbation theory implies that the evolution of each Fourier mode is independent from the other, i.e. we don t have density contrasts of different modes in the equation and their time evolution does not mix.

6 Pressure term The previous equation holds for a pressure-less fluid (w=0). If we consider pressure, it is possible to show that the equation modifies to: New term due to fluid pressure. k is in comoving coordinates, so at each wavenumber k corresponds a physical scale at time t of: From the above equation we identify the Jeans wavelength:

7 Pressure term Each Fourier mode will therefore evolve with time in a different way if the corresponding k is larger or smaller than the Jeans wavelength. If We can neglect this. and we have the growth as discussed for a fluid with w=0.

8 Pressure term If, on the contrary, we are below the Jeans lenght: We neglect the first term and the solution is given by a more complicate oscillation term damped in time.

9 Horizon Scale Given a time t, the quantity: provides a causal horizon, i.e. particles that are a distances larger than it are not causally connected. What happens if I consider a perturbation on scales larger than the horizon scale? We can treat them only using general relativity. As we will see, the solution is also gauge-dependent.

10 Horizon Scale Assuming a synchronous gauge, it is possible to show that perturbations on scales larger than the horizon, i.e. such that at a given time t have: they always grow, as:

11 Summary (single fluid) We have therefore two important scales for structure formation: the horizon scale and the Jeans scale. For cold dark matter (w=0) what is important is the horizon scale at equivalence. Perturbations that enter the horizon before the epoch of equivalence are damped respect to perturbations that enter the horizon later. For CDM the Jeans scale is always zero. For baryons, the Jeans length is approximately the horizon scale until decoupling. The crucial scale is the horizon scale at decoupling. After decoupling baryons have w=0 (approximately). Perturbations that enter the horizon before decoupling (z=1100) are strongly damped respect to perturbations that enter the horizon later.

12 Horizon at equivalence The redshift of equivalence is given by: The Hubble parameter at equivalence is then The physical size is then: And, in comoving coordinates:

13 For example, we can consider two modes, one entering the horizon before the matter-radiation equivalence and another one entering after it. log δ k > k 2 1 Cold dark matter k 1 k 2 Evolution for a w=0 (no pressure) component. (only) Perturbations with k larger than k 1 / a2 = c H ( 2) 2 a a 2 Perturbation in Red: enters the horizon AFTER the equivalence. Perturbation in Blue: enters the horizon BEFORE equivalence. a EQ a 1 log a are damped respect to perturbations with k smaller

14 For example, we can consider two modes, one entering the horizon before the decoupling and another one entering after it. log δ k > k 2 1 Baryons (only) We are below the Jens lenght. Damping+Oscillations k 1 Evolution for the baryon component. Perturbations with k larger than k 1 / a2 = c H ( 2) 2 a a 2 Perturbation in Red: enters the horizon AFTER the decoupling. Perturbation in Blue: enters the horizon BEFORE decoupling. a 1 k 2 log a are strongly damped respect to perturbations with k smaller

15 The situation is different if we consider a CDM+Baryon case. Baryons feel" the CDM gravitational potential. k 1 log δ CDM k 2 +Baryons Baryons, k 2 after decoupling fall in the CDM potential wells. k 1 / a2 = c H ( 2) 2 a a 2 a EQ a 1 log a Baryon/CDM Perturbation in Red: enters the horizon AFTER the decoupling. Baryon Perturbation in Blue: enters the horizon BEFORE equivalence. CDM Perturbation in Green: enters the horizon BEFORE equivalence.

16 Cosmological «Circuit» Generator of Perturbations (Inflation) Amplifier (Gravity) Low band pass filter. Cosmological and Astrophysical effects Tend to erase small scale (large k) perturbations

17 Power Spectrum Each Fourier component is a complex number, which can be written in the form The mean square amplitude of the Fourier components defines the power spectrum : where the average is taken over all possible orientations of the wavenumber. If δ( r) is isotropic, then no information is lost, statistically speaking, if we average the power spectrum over all angles and we get an isotropic power spectrum:

18 Correlation function Let us consider the autocorrelation function of the density field (usually called the correlation function): Where the brackets indicates an average over a volume V. We can write: and, performing the integral we have:

19 Correlation function Since the correlation function is a real number, assuming an isotropic power spectrum we have: If the density field is gaussian, we have that the value of δ at a randomly selected point is drawn from the Gaussian probability distribution: where the standard deviation σ can be computed from the power spectrum:

20 Summary In practice, our theory cannot predict the exact value of in a region of the sky. But if we assume that the initial perturbations are gaussian we can predict the correlation function, the variance of the fluctuations and its the power spectrum P(k). These are things that we can measure using, for example, galaxy surveys and assuming that galaxies trace the CDM distribution.

21 Cosmological «Circuit» Generator of Perturbations (Inflation) Amplifier (Gravity) Low band pass filter. Cosmological and Astrophysical effects Tend to erase small scale (large k) perturbations

22 Power spectrum for CDM The analysis we have presented up to now is very qualitative and approximated (just to have an idea ) The true power spectrum for CDM density fluctuations can be computed by integrating a system of differential equations. We will see this in better detail in the next lectures. In any case, we assume a power law as initial power spectrum as: The motivation of using this type of primordial spectrum comes from inflation (we will discuss this also in a future lecture). We have two free parameters: the amplitude A and the spectral index ns.

23 Power spectrum for CDM The first numerical results on the CDM power spectrum appeared around Bardeen, Bond, Kaiser and Szalay (BBKS) in 1986 proposed the following fitting function: where: This is correct fit for a pure CDM model (no baryons or massive neutrinos). Note the dependence on that defines the epoch of equivalence.

24 P(k) for LCDM (from numerical computations). Spectral index is assumed n=0.96 Primordial regime The position of the peak is related to size of the horizon at equivalence, i.e. to the matter density since radiation is fixed. Note these oscillations in the CDM P(k). Gravitational feedback from baryons. Damping (scales that entered horizon before quality) If we plot the P(k) in function of h/mpc, then the dependence is just on

25 Cold dark matter Effect of the Cosmological parameters

26 Cold dark matter Effect of the Cosmological parameters

27 Baryon density Effect of the Cosmological parameters

28 Baryon density Effect of the Cosmological parameters

29 Spectral index Effect of the Cosmological parameters

30 Spectral index Effect of the Cosmological parameters

31 CDM vs data (1996) If we assume a flat universe made just of matter with we get too much power on small scales to match observations. Already in 1996 the best fit was for i.e. suggesting a low matter density universe.

32 CDM vs data (2003) The 2dfGRS provided the following constraints: (assuming just CDM, no massive neutrinos) 2 sigma detection of baryons:

33 Measurements from SDSS

34 Massive Neutrinos We saw that massive neutrinos contribute to the matter density when they are non-relativistic as: However, until they are relativistic, neutrino have w=1/3, i.e. perturbations in the neutrino component dissipates when they enter the horizon. The wavelength of the horizon when they start to be non relativistic is given by:

35 Numerical evolution of density fluctuations Model with CDM, neutrinos, massive neutrinos, photons, baryons Evolution of the density fields in the synchronous gauge (top panels) and the conformal Newtonian gauge (bottom panels) in the Ων = 0.2 CDM+HDM model for 3 wavenumbers k= 0.01, 0.1 and 1.0 Mpc 1. In each figure, the five lines represent δc, δb, δγ, δν and δh for the CDM (solid), baryon (dashdotted), photon (long-dashed), massless neutrino (dotted), and massive neutrino (short-dashed) components, respectively.

36 Focus just on the Top panel (Syncronous Gauge) CDM, Baryons and MN grow as a^2 and a These fluctuations enter the horizon just after decoupling Photons and massless neutrinos oscillates and damp after entering the horizon

37 Focus just on the Top panel (Syncronous Gauge) These fluctuations enter the horizon just after equality Photons and massless neutrinos oscillates and damp after entering the horizon CDM grow as a^2, ln aˆ2 and a Massive neutrinos (nearly 1.5 ev in this case) are semi relativistic at equality and damped. They slowly catch CDM afterwards. Until decoupling baryons oscillates and are damped, they quickly follow CDM afterwards.

38 Focus just on the Top panel (Syncronous Gauge) These fluctuations enter the horizon well before equality Photons and massless neutrinos oscillates and damp after entering the horizon CDM grow as a^2, ln aˆ2 and a small feedback from b and mnu Massive neutrinos are first relativistic and strongly damped because of free streaming. They try to catch CDM afterwards. Until decoupling baryons oscillates and are damped, they quickly follow CDM afterwards.

39 Baryons vs Massive Neutrinos If a perturbation in baryons enters the horizon before decoupling oscillates with a decreasing amplitude because of diffusion damping (photons can go from hotter to colder regions after several scattering). After decoupling, baryons fall in the CDM potential well. If a perturbation in massive neutrinos enter the horizon before the non relativistic regime is strongly damped because of free streaming (neutrinos are collision-less). Neutrino are very light and also afterwards they still suffer from free streaming and practically don t cluster.

40 Massive Neutrinos The growth of the fluctuations is therefore suppressed on all scales below the horizon when the neutrinos become nonrelativistic the small scale suppression is given by: eff Hu et al, arxiv:astro-ph/ Larger is the neutrino mass, large is the suppression.

41 Massive Neutrinos

42 Massive Neutrinos

43 ...but we have degeneracies... Lowering the matter density suppresses the power spectrum This is virtually degenerate with non-zero neutrino mass

44

45 Inclusion of CMB data is important to break degeneracies Conservative limit from CMB+P(k) measurements:

46 Mass fluctuations Given a theoretical model a quantity can be often easily compared with observations is the variance of fluctuations on a sphere of R Mpc: where: Usually one assumes R=8 Mpc hˆ-1, where the linear approximation is valid.

47 P(k) for LCDM (from numerical computations). Spectral index is assumed n=0.96 The gives the P(k) amplitude around these scales

48 Examples from CAMB

49 Output from CAMB

50 Output from CAMB

51 Output from CAMB

52 Output from CAMB