Origin and Evolution of the Universe Structure Formation

Astro/Phys 224 Spring 2014 Origin and Evolution of the Universe

Joel Primack University of California, Santa Cruz Physics 224 - Spring 2014 – Tentative Term Project Topics Adam Coogan – The BICEP2 Result and Its Implications for Inlationary Cosmology Devon Hollowood – Some Aspect of the Dark Energy Survey Caitlin Johnson – Observing Plan for ACTs to Constrain the Long-Wavelength EBL Thomas Kelly – Various Methods of Testing Modiied Newtonian Dynamics (MOND) Tanmai Sai – Shapes of Simulated Galaxies Vivian Tang – Studying Galaxy Evolution by Comparing Simulations and Observations Determination of σ8 and ΩM from CMB+ WMAP9+SN+Clusters Planck+WP+HighL+BAO

WP = WMAP l<23 Polarization Data Cluster Observations HighL= ACT + SPT

Planck ● ●WMAP9 WMAP7 WMAP5 ● ● ●

BOSS Power Spectrum Normalization Spectrum Power

Cosmic Dark Matter Fraction Planck Clusters vs. CMB?

Planck 2013 results. XX. Cosmology from Sunyaev–Zeldovich cluster counts - arXiv:1303.5080 Assuming a bias between the X-ray determined mass and the true mass of 20%, motivated by comparison of the Planck cluster masses 3 observed mass scaling relations to those from a set of numerical simulations, we ﬁnd that ... σ8 = 0.77 ± 0.02 and The systematic uncertainty quoted here expresses the systematic Ωm = 0.29 ± 0.02. The values of the cosmological parameters are degenerate with the massuncertainty bias, on and the weak-lensing it is masses,found i.e. it includesthat all entries in Table 4 of Applegate et al. (2014) with the exception of the scat- the larger values of σ8 and Ωm preferred by the Planck’s measurements of the primary CMBter anisotropies due to triaxiality, which is accountedcan be for here in the statistical uncertainty. Extending the sample to all 38 clusters yields a consis- accommodated by a mass bias of about 45%. Alternatively, consistency with the primary CMBtent result: constraints can be

+0.039 all = 0.698 0.037 (stat) 0.049 (syst) . achieved by inclusion of processes that suppress power on small scales, such as a component of massive neutrinos.± The weak-lensing masses are expected to yield the true cluster mass on average, and thereby enable a robust calibration of other mass proxies (see discussion in von der Linden et al. 2014; Apple- This has led to papers proposing ∑mν > 0.23 eV such as Hamann & Hasenkampgate etJCAP al. 2014). Therefore,2013 by identifying = (1 b), these results suggest that the mass calibration adopted by the Planck team, (1 b) 0.8, underestimates the true cluster masses by between 5 ⌘ Relative cluster masses can be determined accurately cluster-by-clusterand 25using per cent on average. X-rays as was

done by Vikhlinin+09, Mantz+10, and Planck paper XX, but the absolute masses should3.2 Evidence be for a mass-dependentcalibrated calibration problem using gravitational lensing, say Rozo+13,14 and van der Linden+14. The Arnaud+07,10By eye, Fig. X-ray 1 suggests thatcluster the ratio between masses the WtG weak- lensing and Planck mass estimates depends on the cluster mass: at masses . 6 1014 M , the mass estimates roughly agree, whereas ⇥ used in Planck paper XX are the lowest of all. Using gravitational lensing mass calibrationthe discrepancy appears significant raises for more massivethe clusters. To Figure 1. The ratio of cluster masses measured by Planck and by WtG, for quantify the evidence for such a mass-dependent bias, we use the cluster masses and thus predicts fewer expectedthe clusters commonclusters. to both projects. Solid symbolsThis denote clusterslessens which Bayesian the linear regressiontension method developed between by Kelly (2007) to fit were included in the Planck cluster cosmology analysis (22 clusters) and log(MWtG) as function of log(MPlanck) (fitting the masses directly open symbols additional clusters in the Planck cluster catalog (16 clusters). avoids the correlated errors in the mass ratios one would have to the CMB and cluster observations. The red, solid line indicates a ratio of unity (no bias). The dashed red line account for if fitting the data as shown in Fig. 1). While we show indicates (1 b) = 0.8, the default value assumed throughout most of P16. MPlanck as function of MWtG in Fig. 2 to reflect that the weak-lensing The blue line and shaded regions show our best-fit mass ratio along with masses are our proxy for true cluster masses, we assign the Planck the 1- and 2- confidence intervals. Since the weak-lensing masses are mass estimates to be the independent variable to reduce the e↵ects Closing the loop: self-consistent ... scaling relationsexpected to be unbiased foron average clusters, the ratio of Planck masses of to weak- Galaxies - Rozo+14 MNRAS lensing masses is a measure of the bias (1 b) = M /M of the of Malmquist bias: MPlanck scales with the survey observable, and Planck true Planck cluster masses as used in P20. by choosing it as the independent variable, we provide a mass esti- Robust weak-lensing mass calibration of Planck galaxy cluster massesmate - forvon each datader point Linden+14 which is to first order independent of other Planck cluster masses 3 selection e↵ects (as X-ray selection to first order does not correlate with SZ selection biases, and the lensing data are a subsample of MPlanck/MWtG vs. MPlanck MPlanckan X-ray selectedvs. M catalog).WtG The Kelly (2007) method accounts for The systematic uncertainty quoted here expresses the systematic measurement errors in both variables, as well as for intrinsic scat- uncertainty on the weak-lensing masses, i.e. it includes all entries ter in the dependent variable. Rephrasing the results as a power-law, The ratio of in Table 4 of Applegate et al. (2014) with the exception of the scat- the best-fit relation for the 22 clusters in the cosmology sample is: Noter due Bias to triaxiality, which is accounted for here in the statistical Best Fit cluster 0.76+0.39 uncertainty. Extending the sample to all 38 clusters yields a consis- 0.20 MPlanck +0.077 MWtG tent result: = 0.697 0.095 , 1015 M ⇥ 1015 M masses ! ! +0.039 ⇣ ⌘ all = 0.698 0.037 (stat) 0.049 (syst) . where theThis systematic decreases uncertainty on the weak-lensing the mass cali- ± measured by The weak-lensing masses are expected to yield the true cluster bration is accounted for in the uncertainty on the coecient. In 24 mass on average, and thereby enable a robust calibration of other per centtension of the Monte Carlo samples, between the slope (of log(M Planck) vs. Planck and by mass proxies (see discussion in von der Linden et al. 2014; Apple- log(MWtG)) is unity or larger; i.e. the evidence for a mass-dependent gate et al. 2014). Therefore, by identifying = (1 b), these re- bias is at the 1 level for these 22 clusters. Best Fit clusters⇠ and CMB

Weighing the sults suggest that the mass calibration adopted by the Planck team, To further test for a mass-dependent bias, it is instructive to include the additional 16 clusters in common between Planck and (1 b) 0.8, underestimates⦿ the true cluster masses by between 5 ⌘ Giants (WtG). and 25 per cent on average. WtG thatand are not used the in the Planck motivationcluster cosmology analysis, as

/M these slightly extend the mass range probed. For all 38 clusters, we ﬁnd a consistentfor and∑ morem preciseν > result: 0.23 eV. 3.2 Evidence for500 a mass-dependent calibration problem . +0.15 0 68 0.11

MPlanck +0.059 MWtG Planck By eye, Fig. 1 suggestsM that the ratio between the WtG weak- = 0.699 0.060 . 1015 M ⇥ 1015 M lensing and Planck mass estimates depends MText on500 the/M cluster Text⦿ mass:WtG ! ⇣ ⌘ ! at masses . 6 1014 M , the mass estimates roughly agree, whereas In 4.9 per cent of the Monte Carlo samples, the slope is unity or ⇥ Figure 2. The direct comparison between M500 cluster masses measured by larger; i.e. the confidence level for a slope less than unity is 95 per the discrepancyPlanck appearsand significant by WtG. The for symbols more massive are the same clusters. as in Fig. To 1. The green line cent3. Figure 1. The ratio of cluster masses measured by Planck and by WtG, for quantify the evidenceand shaded for such regions a mass-dependentshow the best-fit linear bias, relation we use between the the logarithmic the clusters common to both projects. Solid symbols denote clusters which Bayesian linear regressionmasses and methodits 1- and developed 2- confidence by Kelly intervals (2007) (the to fit fit was performed with were included in the Planck cluster cosmology analysis (22 clusters) and log(MWtG) as functionlog(MWtG of) as log( functionMPlanck of) log( (fittingMPlanck the)). masses directly 3 We note that when using bootstrap realizations of an unweighted simple open symbols additional clusters in the Planck cluster catalog (16 clusters). avoids the correlated errors in the mass ratios one would have to linear regression as a more agnostic fit statistic, we recover the same slope, The red, solid line indicates a ratio of unity (no bias). The dashed red line account for if fitting the data as shown in Fig. 1). While we show indicates (1 b) = 0.8, the default value assumed throughout most of P16. MPlanck as functionc 2014 of M RAS,WtG in MNRAS Fig. 2 to000 reflect, 1–5 that the weak-lensing The blue line and shaded regions show our best-fit mass ratio along with masses are our proxy for true cluster masses, we assign the Planck the 1- and 2- confidence intervals. Since the weak-lensing masses are mass estimates to be the independent variable to reduce the e↵ects expected to be unbiased on average, the ratio of Planck masses to weak- lensing masses is a measure of the bias (1 b) = M /M of the of Malmquist bias: MPlanck scales with the survey observable, and Planck true Planck cluster masses as used in P20. by choosing it as the independent variable, we provide a mass esti- mate for each data point which is to first order independent of other selection e↵ects (as X-ray selection to first order does not correlate with SZ selection biases, and the lensing data are a subsample of an X-ray selected catalog). The Kelly (2007) method accounts for measurement errors in both variables, as well as for intrinsic scat- ter in the dependent variable. Rephrasing the results as a power-law, the best-fit relation for the 22 clusters in the cosmology sample is:

. +0.39 0 76 0.20 MPlanck +0.077 MWtG = 0.697 0.095 , 1015 M ⇥ 1015 M ! ⇣ ⌘ ! where the systematic uncertainty on the weak-lensing mass calibration is accounted for in the uncertainty on the coecient. In 24 per cent of the Monte Carlo samples, the slope (of log(MPlanck) vs. log(MWtG)) is unity or larger; i.e. the evidence for a mass-dependent bias is at the 1 level for these 22 clusters. ⇠ To further test for a mass-dependent bias, it is instructive to include the additional 16 clusters in common between Planck and WtG that are not used in the Planck cluster cosmology analysis, as these slightly extend the mass range probed. For all 38 clusters, we ﬁnd a consistent and more precise result:

. +0.15 0 68 0.11 MPlanck +0.059 MWtG = 0.699 0.060 . 1015 M ⇥ 1015 M ! ⇣ ⌘ ! In 4.9 per cent of the Monte Carlo samples, the slope is unity or Figure 2. The direct comparison between M500 cluster masses measured by larger; i.e. the confidence level for a slope less than unity is 95 per Planck and by WtG. The symbols are the same as in Fig. 1. The green line cent3. and shaded regions show the best-fit linear relation between the logarithmic masses and its 1- and 2- confidence intervals (the fit was performed with log(MWtG) as function of log(MPlanck)). 3 We note that when using bootstrap realizations of an unweighted simple linear regression as a more agnostic fit statistic, we recover the same slope,

c 2014 RAS, MNRAS 000, 1–5 Late Cosmological Epochs

380 kyr z~1000 recombination last scattering

dark ages

~100 Myr z~30 first stars ~480 Myr z~10 “reionization”

galaxy formation

13.7 Gyr z=0 today 6 FLUCTUATIONS: LINEAR THEORY Recall: (here a = R, Λ=0) “TOP“TOP HAT”HAT MODEL”MODEL

7 GROWING MODE 8 The Initial Fluctuations

At Inflation: Gaussian, adiabatic

! ! ! δ (x) = δ ! eik⋅x Fourier transform: ∑! k k ~ ! Power Spectrum: P(k) ≡ <| δ (k ) |2 > ∝ k n rms perturbation: Gravitational Instability: Dark Matter

Small fluctuations: comoving coordinates

Continuity: matter era Euler: Poisson:

Linear approximation:

growing mode:

irrotational, potential flow: 8PieroMadau:Early Galaxy Formation

8PieroMadau:Early Galaxy Formation

Figure 3. Evolution of the radiation (dashed line,labeledCMB)andmatter(solid line, labeled GAS) temperatures after recombination, in the absence of any reheating mechanism.

The coefficient of the fractional temperature difference reaches unity at the “thermaliza- tion redshift” z 130. That is, the residual ionization is enough to keep the matter in th ≈The coefficient of the fractional temperature difference reaches unity at the “thermaliza- temperature equilibriumtion redshift” with thez CMB130. well That after is, the decoupling residual ionization.Atredshiftlowerthan is enough to keep thez math tter in th ≈ the temperature oftemperature intergalactic equilibrium gas falls with adiabatically the CMB well faster after than decoupling that.Atredshiftlowerthan of the radiation, zth −2 Te a .Fromtheanalysisabove,therateofchangeoftheradiation the Thus temperature far, we have of considered intergalactic only gasthe evolution falls adiabatically of fluctuations faster in the than energydark that matter. of density the But radiation, of ∝ course we −have2 to consider also the ordinary matter, known in cosmology as due to Compton scatteringTe a .Fromtheanalysisabove,therateofchangeoftheradiation can be written as energy density “baryons”due∝ to Compton(implicitly scatteringincluding the can electrons). be written See as Madau’s lectures “The Astrophysics of Early Galaxy Formation” (http://arxiv.org/abs/0706.0123v14 ) for a summary. We have already seen du 4 σT aBT 3kBne 4 that the baryons are= primarily in thedu form4 ofσ Tatoms(aTBeT after3Tk)B ,zn ~e 1000, with a residual ionization(2.29) fraction of a fewdt x 10-43. Theyc become= mfullye reionized− by z ~( T6,e butT they), were not reionized at (2.29) dt 3 c me − z~20 since the COBE satellite found that “Compton parameter” y ≤ 1.5 x 10-5, where or or Figure 3. du du kB(Te T ) kB(Te kBT()Te Tk)B(T2 e 2T2 ) Evolution of the radiation (dashed line,labeledCMB)andmatter(solid line, =4dy,with dy (n σ cdt) −, σT= = d(8τπ/3)(e /mc− ) . (2.30) =4dy, dy (neσT cdt) e −2T = dτe 2 −2 labelede . GAS)2 (2.30) temperatures after recombination, in the absence of any reheating mechanism. u u≡ ≡ mec mec mec mec Compton scatteringThisCompton causes implies a scatteringthat distorsion causes of the a distorsion CMB spectrum, of the CMBThus,depopulating spectrum, for example,depopulating a the universe Rayleigh- that the Rayleigh- Jeans regime in favor of photons in the Wien tail. The5 “Compton-parameter” Jeans regime in favorwas reionized of photons and reheated in the at Wien z = 20 tail.to (xe, TheTe) = (1, “Compto > 4×10 K)n-parameter” would violate the COBE y-limit. z k T dτ z y = B e e dz (2.31) kB Te dτe 2 The figure at right shows the !0 mec dz The coefficient of the fractional temperature difference reaches unity at the “thermaliza- y = 2 dz (2.31) evolution of the radiation!0 (dashedmec dz tion redshift” zth 130. That is, the residual ionization is enough to keep the matter in is aline, dimensionless labeled CMB measure) and matter of this distorsion, and is proportional to the pressure≈ of the temperature equilibrium with the CMB well after decoupling.Atredshiftlowerthanzth is a dimensionless measureelectron(solid gasline, ofn thislabeledekBT distorsion,e .TheGAS) COBE andsatellite is propor hastional shown the to the CMB pressure to be thermal of the to high −5 the temperature of intergalactic gas falls adiabatically faster than that of the radiation, accuracy,temperatures setting after a limit recombination,y 1.5 10in (Fixsen et al. 1996). This can−2 be shown to imply electron gas nekBTe.TheCOBE satellite≤ has× shown the CMB to beTe thermala .Fromtheanalysisabove,therateofchangeoftheradiation to high energy density the absence of any −reheating5 3/2 7 ∝ accuracy, setting a limit y 1.5 10 (FixsenxeTe [(1 et + al.z) 1996).1] This< 4 can10 K bedue. shown to Compton to imply scattering(2.32) can be written as mechanism.≤ × ⟨ ⟩ − × (From Madau’s lectures, at Auniversethatwasreionizedandreheatedat3/2 7 z =20to(x ,T )=(1,>4 105 K), for 4 physics.ucsc.edu/~joel/Phys224xeTe [(1 + z) 1] .) < 4 10 K. e e (2.32) du 4 σT aBT 3kBne example,⟨ would⟩ violate the COBE− y-limit.× × = (Te T ), (2.29) dt 3 c me − 5 Auniversethatwasreionizedandreheatedatz =20to(xe,Te)=(1,>or 4 10 K), for example, would violate the COBE y-limit. × du kB(Te T ) kB(Te T ) =4dy, dy (neσT cdt) −2 = dτe −2 . (2.30) u ≡ mec mec Compton scattering causes a distorsion of the CMB spectrum, depopulating the Rayleigh- Jeans regime in favor of photons in the Wien tail. The “Compton-parameter” z kB Te dτe y = 2 dz (2.31) !0 mec dz is a dimensionless measure of this distorsion, and is proportional to the pressure of the electron gas nekBTe.TheCOBE satellite has shown the CMB to be thermal to high accuracy, setting a limit y 1.5 10−5 (Fixsen et al. 1996). This can be shown to imply ≤ × x T [(1 + z)3/2 1] < 4 107 K. (2.32) ⟨ e e⟩ − × 5 Auniversethatwasreionizedandreheatedatz =20to(xe,Te)=(1,>4 10 K), for example, would violate the COBE y-limit. × The linear evolution of sub-horizon density perturbations in the dark matter-baryon fluid is governed in the matter-dominated era by two second-order differential equations:

(1) for the dark matter, and “Hubble friction”

for the baryons, where δdm(k) and δb(k) are the Fourier components of the density ﬂuctuations in the dark matter and baryons,† fdm and fb are the corresponding mass fractions, cs is the gas sound speed, k is the (comoving) wavenumber, and the derivatives are taken with respect to cosmic time. Here

(2)

is the time-dependent matter density parameter, and ρ(t) is the total background matter density. Because there is ~5 times more dark matter than baryons, it is the former that deﬁnes the pattern of gravitational wells in which structure formation occurs. In the case where fb ≃ 0 and the universe is static (H = 0), equation (1) above becomes

† For each fluid component (i = b, dm) the real space fluctuation in the density field, can be written as a sum over Fourier modes, where tdyn denotes the dynamical timescale. This equation has the solution

After a few dynamical times, only the exponentially growing term is significant: gravity tends to make small density fluctuations in a static pressureless medium grow exponentially with time. Sir James Jeans (1902) was the first to discuss this. The additional term ∝ H present in an expanding universe can be thought as a “Hubble friction” term that acts to slow down the growth of density perturbations. Equation (1) admits the general solution for the growing mode:

(3) so that an Einstein-de Sitter universe gives the familiar scaling δdm(a) = a with coefﬁcient unity. The right-hand side of equation (3) is called the linear growth factor D(a) = D+(a).

Different values of Ωm, ΩΛ lead to different linear growth factors. Growing modes actually decrease in density, but not as fast as the average universe. Note how, in contrast to the exponential growth found in the static case, the growth of perturbations even in the case of an Einstein-de Sitter (Ωm =1) universe is just algebraic rather than exponential. This was discovered by the Russian physicist Lifshitz (1946).

The consequence is that dark matter fluctuations grow proportionally to the scale factor a(t) when matter is the dominant component of the universe, but only logarithmically when radiation is dominant. Thus there is not much inside horizon difference in the amplitudes of fluctuations outside horizon of mass M < 1015 Msun, which enter the horizon before zmr ~ 4 ×103, while there is a stronger dependance on M for fluctuations with M > 1015 Msun.

Primack & Blumenthal 1983

There is a similar suppression of the growth of matter ﬂuctuations once the gravitationally dominant component of the universe is the dark energy, for example a cosmological constant. Lahav, Lilje, Primack, & Rees (1991) showed that the growth factor in this case is well approximated by

Here is again given by The Linear Transfer Function T(k) An approximate ﬁtting function for T(k) in a ΛCDM universe is (Bardeen et al. 1986)

where (Sugayama 1995)

For accurate work, for example for starting high-resolution N-body simulations, it is best to use instead of ﬁtting functions the numerical output of highly accurate integration of the Boltzmann equations, for example from CMBFast, which is available at http://lambda.gsfc.nasa.gov/toolbox/ which points to http://lambda.gsfc.nasa.gov/toolbox/tb_cmbfast_ov.cfm

W e l c o m e to the CMBFAST Website! This is the most extensively used code for computing cosmic microwave background anisotropy, polarization and matter power spectra. The code has been tested over a wide range of cosmological parameters. We are continuously testing and updating the code based on suggestions from the cosmological community. Do not hesitate to contact us if you have any questions or suggestions. U. Seljak & M. Zaldarriaga

Scale-Invariant Spectrum (Harrison-Zel’dovich) mass

t time

M CDM Power Spectrum

mass

δ growth when matter is self-gravitating

teq t time

CDM δ Pk

HDM HDM CDM free streaming

Meq mass kpeak ∝ Ωm h k Formation of Large-Scale Structure

Fluctuation growth in the linear regime:

rms fluctuation at mass scale M:

CDM: bottom-up HDM: top-down

1 1

free streaming 0 0 M M

Galaxies Clusters Superclusters Galaxies Clusters Superclusters Einstein-de Sitter Structure forms Open universe earliest in Open, next in Benchmark, Open Benchmark model latest in EdS model. Benchmark

EdS

From Peter Schneider, Extragalactic Astronomy and Cosmology (Springer, 2006) Linear Growth Rate Function D(a)

From Klypin, Trujillo-Gomez, Primack - Bolshoi paper 1 (2011, ApJ 740, 102) - Appendix A P(k) nonlinear (σ8, Γ) linear

From Peter Schneider, Extragalactic Astronomy and Cosmology (Springer, 2006)

. The shape parameter Γ = Ωmh. On large scales (k small), the gravity of the dark matter dominates. But on small scales, pressure dominates and growth of baryonic fluctuations is prevented. Gravity and pressure are equal at the Jeans scale

The Jeans mass is the dark matter + baryon mass enclosed within a sphere of radius πa/kJ,

where µ is the mean molecular weight. The evolution of MJ is shown below, assuming that reionization occurs at z=15: Hot Dark Matter

Cold Dark Matter

Warm Dark Matter

Jeans-type analysis for HDM, WDM, and CDM GRAVITY – The Ultimate Capitalist Principle Astronomers say that a region of the universe with more matter is “richer.” Gravity magnifies differences—if one region is slightly denser than average, it will expand slightly more slowly and grow relatively denser than its surroundings, while regions with less than average density will become increasingly less dense. The rich always get richer, and the poor poorer.

The early universe expands almost perfectly uniformly. But there are small differences in density from place to place (about 30 parts per million). Because of gravity, denser regions expand more slowly, less dense regions more rapidly. Thus gravity amplifies the contrast between them, until… Temperature map at 380,000 years after the Big Bang. Blue (cooler) regions are slightly denser. From NASA’s WMAP satellite, 2003. Structure Formation by Gravitational Collapse

When any region and starts falling Through Violent becomes about together. The forces Relaxation the dark twice as dense as between the matter quickly reaches a typical regions its subregions generate stable configuration size, it reaches a velocities which that’s about half the maximum radius, prevent the material maximum radius but stops expanding, from all falling denser in the center. toward the center. Simulation of top-hat collapse: Used in my 1984 summer school lectures “Dark matter, Galaxies, P.J.E. Peebles 1970, ApJ, 75, 13. and Large Scale Structure,” http://tinyurl.com/3bjknb3 rrv rrmaxm vir

TOP HAT VIOLENT VIRIALIZED Max Expansion RELAXATION Growth and Collapse of Fluctuations

Schematic sketches of radius, density, and density contrast of an overdense fluctuation. It initially expands with the Hubble expansion, reaches a maximum radius (solid vertical line), and undergoes violent relaxation during collapse (dashed vertical line), which results in the dissipationless matter forming a stable halo. Meanwhile the ordinary matter ρb continues to dissipate kinetic energy and contract, thereby becoming more tightly bound, until dissipation is halted by star or disk formation, explaining the origin of galactic spheroids and disks. (This was the simplified discussion of BFPR84; the figure is from my 1984 lectures at the Varenna school. Now we take into account halo growth by accretion, and the usual assumption is that large stellar spheroids form mostly as a result of galaxy mergers Toomre 1977. But now we think that the most intermediate mass stellar spheroids form because of disk instability.)