Dynamics of Structure Formation

Transcription

1 Overview Dyamics of Structure Formatio Physics of desity perturbatio evolutio Dyamics of liear perturbatios The emergece of structures over a broad rage of scales from a highly homogeeous early Uiverse is oe of the key areas of study i cosmology. Some simple tools have bee developed to describe the evolutio of desity perturbatios i the liear regime Power Spectra ad Trasfer fuctios 4. May 018 Frotiers of Cosmology - Mohr - Lecture Our uiverse the ad ow Large Scale Structure Recombiatio (~400,000 yr) dr/<r> ~ 10-5 Cosmic Backgroud Explorer (NASA) Preset (~14x10 9 yr) dr/<r> ~ 10 6 Harvard-Smithsoia Ceter for Astrophysics Las Campaas Redshift Survey 4. May 018 Frotiers of Cosmology - Mohr - Lecture 3 4. May 018 Frotiers of Cosmology - Mohr - Lecture 4

2 Desity perturbatios Study of the evolutio of desity perturbatios is cast i terms of the evolutio of the dimesioless (eergy) desity perturbatio d 1+δ( x ) ρ ( x ) ρ Desity cotet is divided ito o-relativistic matter ad radiatio (relativistic matter) Adiabatic perturbatios: variatios i umber desity that affect all species by the same factor. Imagie small compressios or expasios. This implies differet eergy desity variatios i the two species δ r = 4 3 δ m Isocurvature perturbatios: etropy perturbatios: where the total desity remais costat, so ρ r δ r = ρ m δ m At times early compared to matter-radiatio equality, these perturbatios correspod to variatios i the matter desity that are offset by miiscule perturbatios i the radiatio desity 4. May 018 Frotiers of Cosmology - Mohr - Lecture 5 Adiabatic versus Isocurvature Isocurvature desity perturbatios seem to be more atural, because causality makes it impossible to chage desity o scales larger tha the horizo These perturbatios are seeded i late time cosmological phase trasitio models Withi iflatioary models the particle horizo is chaged at early times so that total desity fluctuatios ca be imprited o scales that appear to be larger tha the horizo. If curvature fluctuatios are imprited prior to the process resposible for baryo asymmetry the adiabatic modes are orm See discussio i Chapter 11.5, Peacock Observatioal evidece of adiabatic desity fluctuatios the provide good support for iflatioary models. For example, CMB aisotropy studies allow oe to costrai the mix of adiabatic ad isocurvature fluctuatios. 4. May 018 Frotiers of Cosmology - Mohr - Lecture 6 Matter ad Trasfer Fuctios Pressure Effects: Jeas Istability Matter cotet affects desity perturbatios through self-gravitatio, pressure ad dissipative processes. Liear adiabatic perturbatios grow as % δ a( t) radiatio domiatio & ' a( t) matter domiatio Ω =1 Isocurvature perturbatios are iitially costat ad the declie & δ m costat radiatio domiatio ' ( a( t) 1 matter domiatio Ω =1 I first case, gravity is workig to icrease the overdesity ad i secod case gravity is workig to maitai the homogeeity. I both cases the shape of the desity perturbatio is uchaged, ad its amplitude is evolvig with time Jeas istability occurs i the collapse of a cloud whe the restorig pressure force is icapable of offsettig the gravitatioal collapse durig a perturbatio Oe ca derive the Jeas legth by comparig the soud wave crossig time ad the gravitatioal free fall timescale i a spherical cloud t s = R c s whereas t ff π Gρ e.g. Scalig follows from Keplers 3 rd Law λ J = c s P = 4π GM R3 π Gρ M J = 4 πλ 3 3 J c 3 s ρ 3 4. May 018 Frotiers of Cosmology - Mohr - Lecture 7 4. May 018 Frotiers of Cosmology - Mohr - Lecture 8

3 Pressure Effects i our Uiverse Small scale effects: Silk dampig Pressure opposes gravity effectively for wavelegths below the Jeas legth While the uiverse is radiatio domiated the soud speed is c π s = c λ J = c s R 3 H = c 3 = c Gρ H 8πGρ ad so the Jeas legth is always close to the size of the horizo. Jeas legth reaches a maximum at matter-radiatio equality ad the soud speed drops ad Jeas legth decreases. Photo diffusio ca erase perturbatios i the matter-radiatio fluid Distace travelled by the photo radom walk by the time of the last scatterig is λ s =.7( Ω m Ω B h 6 ) 1 4 Mpc 15Mpc The comovig Jeas legth at M-R equality is R o r H Ω m z eq ( z eq ) = 1 1 c H o 130Mpc At larger scales, perturbatios should be affected oly by gravity, ad below this scale pressure forces are importat ad growth is slowed or stopped. Because this scale depeds o the matter desity ad because the galaxy distributio reflects the uderlyig distributio of desity perturbatios, oe would expect a measure of the galaxy desity distributio to provide a measure of the matter desity Withi over-desity, probability to leave exceeds probability to arrive Diffusio of photos out of over-dese regios affects baryos, too, because the radiatio ad baryos are tightly coupled prior to recombiatio Models with dark matter are less impacted because dark matter perturbatios remai ad baryos ca fall back ito those potetial wells after last scatterig 4. May 018 Frotiers of Cosmology - Mohr - Lecture 9 4. May 018 Frotiers of Cosmology - Mohr - Lecture 10 Small scale effects: Free streamig Noliear processes At early times dark matter particles will udergo free streamig at the speed of light, erasig all scales up to the horizo Most of the directly observable objects i the Uiverse have already trasitioed well beyod the liear regime, ad this presets a challege This cotiues util the particles go o-relativistic c λ fs = H(z r ) For light massive eutrios (hot dark matter) this happes at z eq, ad essetially all structures o scales smaller tha the horizo at M-R equality are erased. With cold dark matter the particles are much more massive ad go o-relativistic earlier. These differeces lead to dramatically differet structures, ad ideed hot dark matter is ruled out. A measure of the characteristic amplitude of small scale desity fluctuatios prior to their goig o-liear should provide a costrait o the dark matter particle mass (if the dark matter particle is a thermal relic) Equatios of motio ca be itegrated i N-body simulatios, ad extesive work has bee doe o developig aalytical models to describe oliear evolutio But the distributio of fluctuatios i the microwave backgroud, the clusterig of objects o sufficietly large scale, probes of clusterig that exted to the high redshift uiverse ad the umber desity of collapsed massive objects like clusters all are examples of observatios whose iterpretatio relies primarily o liear evolutio of desity perturbatios 4. May 018 Frotiers of Cosmology - Mohr - Lecture May 018 Frotiers of Cosmology - Mohr - Lecture 1

4 Overview Dyamics of liear perturbatios Physics of desity perturbatio evolutio Dyamics of liear perturbatios Oe ca study the evolutio of desity perturbatios i the liear regime withi a Newtoia framework Euler Dv = p dt ρ Φ Power Spectra ad Trasfer fuctios Eergy Poisso Dρ = ρ v dt Φ = 4πGρ Material or total derivative with covective term D dt = t + v 4. May 018 Frotiers of Cosmology - Mohr - Lecture 13 The itroduce perturbatios as r=r o +dr ad v=v o +dv ad collect terms that are first order i the perturbed quatities. Itroduce δ δρ ρ o 4. May 018 Frotiers of Cosmology - Mohr - Lecture 14 First order perturbatio evolutio Comovig coordiates To first order i the perturbed quatities, the goverig equatios become: d dt δ v = δp δφ ( δv ) ρ o d δ = δ v dt δφ = 4πGρ o δ where d/dt is the time derivative of a observer comovig with the uperturbed expasio of the uiverse v o d dt = t + v o Through a trasformatio to comovig coordiates it is possible to describe the evolutio of the perturbed quatities with respect to overall uiform expasio. We itroduce x (t) = a(t) r (t) δv (t) = a(t) u (t) ad use x = 1 a r g = δφ a The dyamical equatios become u + a a u = g a δp ρ o δ = u δφ = 4πGρ o δ 4. May 018 Frotiers of Cosmology - Mohr - Lecture May 018 Frotiers of Cosmology - Mohr - Lecture 16

5 Differetial equatio for perturbatio The o-expadig case Usig a equatio of state defiitio of the soud speed c s p ρ Without the uiform expasio, the differetial equatio is δ = δ 4πGρ o c s k Ad cosiderig a plae wave perturbatio where k is the comovig wavevector we obtai: δ e i k r δ + a & δ = δ 4πGρ a o c s ( ' k a ) + * With solutios of the form δ(t) = e ±t τ where τ =1/ 4πGρ o c s k This solutio uderscores the possibility of pressure stability ad oe ca see the critical Jeas legth l J =p/k J i the expoet which govers the trasitio from real to imagiary t λ J = c s π Gρ 4. May 018 Frotiers of Cosmology - Mohr - Lecture May 018 Frotiers of Cosmology - Mohr - Lecture 18 Evolutio durig radiatio domiatio Soud speed differs ad the overall treatmet we just discussed is iappropriate c s = c 3 The costrait equatios become: Dv = Φ dt D ρ + p c dt = ( p c ) ρ + p c t v Φ = 4πG ρ + 3 p c The differetial equatio describig evolutio of the overdesity becomes (where we have used p = ρc 3 ) δ + a δ = 3π a 3 Gρ oδ 4. May 018 Frotiers of Cosmology - Mohr - Lecture 19 Solutios for d(t) If we try a power law solutio i t we obtai δ(t) t 3 or t 1 matter domiated δ(t) t 1 or t 1 radiatio domiated Remember that for W=1, accordig to the Friedma equatio, the scale factor grows as a(t) t 3 matter domiated a(t) t 1 radiatio domiated Givig us simple solutios for the growth of desity perturbatios for these two cases (early ad itermediate times) δ a matter domiated δ a radiatio domiated As dark eergy comes to domiate at late times the solutio deviates from these simple cases 4. May 018 Frotiers of Cosmology - Mohr - Lecture 0

6 Overview Desity Field ad Pairs Physics of desity perturbatio evolutio Dyamics of liear perturbatios Power Spectra ad Trasfer fuctios The cosmic desity field r is typically expressed i dimesioless form d, where ρ is the mea desity ρ (x) ρ δ (x) = ρ The correlatio fuctio ξ is defied as the overabudace of galaxypairs at separatio r relative to a radom distributio: dn pair r = dn1dn = 0 (1+ ξ ( r ))dv1dv () The expected umber of pairs i two volumes dv1 ad dv separated by r ca also be writte as the volume average of the followig quatity dn pair = 0 (1+ δ ( x))dvx 0 (1+ δ ( x + r ))dvx+r 4. May 018 Frotiers of Cosmology - Mohr - Lecture 1 Correlatio Fuctio ad Overdesity Frotiers of Cosmology - Mohr - Lecture Correlatio Fuctio ad Power Spectrum Averagig over the volume we write () 1 = 0 dv1dv 1+ d 3 x δ ( x)δ ( x + r ) V VolumeV Because The correlatio fuctio is the covolutio of the overdesity field with itself (offset by distace r) so the auto-correlatio fuctio ξ ( r ) = δ ( x)δ ( x + r ) = (δ δ )r 1 dn pair r = d 3 x 0 (1+ δ ( x) + δ ( x + r ) + δ ( x) δ ( x + r ))dv1dv V VolumeV Usig the covolutio theorem, we ca write ξ ( r ) = δ ( x)δ ( x + r ) V (π )3 V = (π )3 d 3 x δ ( x) = 0 = VolumeV By comparig these expressios we see that: ξ ( r ) = δ ( x)δ ( x + r ) The (auto)correlatio fuctio is just the expectatio value of the product of the overdesity field times the field offset by distace r Frotiers of Cosmology - Mohr - Lecture 3 ik r k k δ δ e δ k d 3k where δk = 3 d xδ (x)e ik x e ik r d 3k So the correlatio fuctio is the Fourier Trasform pair of the power spectrum P(k)= dk Because Uiverse is isotropic, drop vector quatities. P (k ) = 1 V ξ (r ) e ikr d 3r ad ξ ( r ) = V ( π ) Frotiers of Cosmology - Mohr - Lecture 3 P (k ) e ikr d 3k 4

7 Power Spectrum Fourier aalyses are particularly coveiet for may applicatios The power spectrum ecodes the secod momet of the desity field (variace, because zero mea). Statistical homogeeity ad isotropy implies P(k) Dimesioless form of power spectrum commoly used. Ecodes probability of desity fluctuatio existig above some threshold (i.e. d c ) as fuctio of k per uit logarithmic iterval dk/k = 1 V d 3 r δ r δ k δ r e i k r = V d 3 k δ k ( π ) 3 P( k 1, k ) = e i k r 1 δ ( k ( π ) 3 1 )δ k P k 1, k ( ) = δ D ( k1 k )P( k 1 ) Δ ( k) = k3 π P k Statistical Descriptio of Desity Perturbatios A reasoable startig assumptio would be a desity field where the phases of differet Fourier modes are ucorrelated ad radom, which would be a Gaussia radom field Withi this cotext the Power Spectrum, which gives the characteristic variace of these fluctuatios as a fuctio of scale, cotais all the iformatio P(k) δ k Gaussiaity to a high level is expected from Iflatio, ad it ca be measured i the CMB ad i the impact that o-gaussiaity would have o structure formatio. We retur to this later. 4. May 018 Frotiers of Cosmology - Mohr - Lecture 5 4. May 018 Frotiers of Cosmology - Mohr - Lecture 6 LSS Costrais Power Spectrum = Gaussia radom field d(x) = Liear power spectrum P(k) Las Campaas Redshift Survey 4. May 018 Frotiers of Cosmology - Mohr - Lecture 7 Power Spectrum ad Trasfer fuctio Real power spectra result from the processig of iitial or primordial power spectra by the physics we ve bee discussig: self-gravitatio, pressure, silk dampig ad free streamig. P(k) δ k Because this physics ad the primordial power spectrum are a fuctio of scale all modes of a particular physical scale are affected similarly it is coveiet to ecapsulate these effects i a trasfer fuctio T k δ k( z = 0) ( z)d(z) δ k = P o k T ( k) where D(z) is the liear growth factor betwee redshift z ad the preset ad k is the comovig waveumber of the mode. The ormalizatio redshift is uimportat as log as it refers to a time before ay scale of iterest has etered the horizo 4. May 018 Frotiers of Cosmology - Mohr - Lecture 8

8 Iitial Power Spectrum Iitial perturbatios imprited durig iflatio Spectral idex determies variatio with k P( k) k ~1 (just below) expected from iflatio models. CMB aisotropy (WMAP) gives =0.968+/-0.01 Δ from slide 5 ( k) = k3 π P k Scale Ivariat Power Spectrum The Harriso-Zeldovich scale ivariat power spectrum has =1 Δ k So i terms of desity fluctuatios a scale ivariat spectrum implies higher characteristic amplitudes for perturbatios o smaller scalesà i other words, it implies bottom up structure formatio Cosider a fluctuatio i the gravitatioal potetial, which scales as d k /k k 3 P(k) k +3 δφ = 4πGρ o δ δφ k = 4πGρ o δ k /k Thus, the HZ spectrum implies scale idepedet potetial (or curvature) fluctuatios Δ Φ ( k) k 4 Δ δ ( k) k 1 P δ (k) k 1 4. May 018 Frotiers of Cosmology - Mohr - Lecture 9 4. May 018 Frotiers of Cosmology - Mohr - Lecture 30 Iflatioary predictios HZ spectrum has a similar amout of power i potetial (curvature) fluctuatios withi logarithmic itervals o all scales. Oe has the same amplitude potetial fluctuatios comig through the horizo at all times I iflatioary models the amplitude of the potetial/curvature fluctuatios is related to the value of the expasio parameter H. The Friedma equatio tells us that expoetial expasio requires costat H, ad therefore over the period of expoetial expasio (iflatio) a scale ivariat spectrum of desity perturbatios should be produced Detailed iflatioary models predict adiabatic perturbatios (potetial/curvature fluctuatios) that are close to Harriso-Zeldovich but with slightly higher gravitatioal potetial fluctuatio amplitudes o larger physical scales (tilt; close to but < 1). This suggests a tedecy for the H parameter to have falle durig iflatio Liear growth ad additioal effects Calculatio of the trasfer fuctios is a techical challege because there is a mixture of matter (dm ad baryos) ad relativistic particles (eutrios, photos) These techical difficulties have bee overcome i publicly available codes like CMBfast ad CAMB. There are two ways for the power spectrum today to differ from that which was laid dow at early times (durig Iflatio): Jeas mass effects- pressure as a balacig force to gravity Dampig effects- overdese regios decay through free streamig or Silk dampig (As growth proceeds it exteds ito the o-liear regime ad this chages also the power spectrum) 4. May 018 Frotiers of Cosmology - Mohr - Lecture May 018 Frotiers of Cosmology - Mohr - Lecture 3

9 Jeas mass effects T(k) for adiabatic perturbatios Prior to M-R equality perturbatios that lie iside the horizo are preveted from growig by radiatio pressure. The horizo scale at M-R equality is d eq = 39 Ωh 1 Mpc 30 Mpc After z eq, perturbatios o all scales grow if collisioless dark matter domiates; however if baryoic gas domiates the the Jeas legth remais approximately costat ad growth suppressio is maitaied through recombiatio For larger scale perturbatios that do t eter the horizo util after z eq they udergo growth δ a λ a d H = c H H = 8πG 3 ρ = 8πG 3 ρ oa 4 d H a For smaller scale perturbatios with waveumber k that eter the horizo prior to z eq, they eter ito a sort of oscillatory stasis due to pressure effects. This period of lost growth suppresses their growth, ad the suppressed growth effect is larger for the smaller perturbatios that eter the horizo earlier The impact o the trasfer fuctio depeds o the type of perturbatio For adiabatic perturbatios this implies: $ & 1 (kd eq <<1) T k % ( kd eq ) '& (kd eq >>1) 4. May 018 Frotiers of Cosmology - Mohr - Lecture May 018 Frotiers of Cosmology - Mohr - Lecture 34 T(k) for isocurvature perturbatios Trasfer Fuctios For smaller scale perturbatios that eter the horizo prior to z eq all the photos disperse (free stream) ad oly the matter perturbatios remai, which the basically remai costat util after z eq Larger scale perturbatios that eter after z eq grow from that poit o (do t grow while outside horizo because remember these perturbatios have zero associated eergy desity perturbatio) So the et effect is a kid of opposite behavior relative to the adiabatic perturbatios. For isocurvature perturbatios this implies: *, T k + -, # $ & ( ' % kc 15 Ho 3 1 a eq (kd eq <<1) (kd eq >>1) This plot from Peacock illustrates the behavior of the trasfer fuctio i several cases, icludig both adiabatic ad isocurvature as well as purely CDM, HDM ad baryoic models P(k) δ k = P o k T ( k) 4. May 018 Frotiers of Cosmology - Mohr - Lecture May 018 Frotiers of Cosmology - Mohr - Lecture 36

10 Measured Power Spectrum I the real Uiverse oe ca costrai the power spectrum i a variety of ways: The temperature perturbatios of the CMB ca be directly coected to the uderlyig baryoic desity perturbatios ad uderlyig matter desity perturbatios Matter Desity Impact o Power Spectrum Durig radiatio domiatio dark matter- radiatio couplig leads to stasis for dark matter desity perturbatios o scales below Jeas scale Δ from slide 5 ( k) = k3 π P k With weak lesig it is possible to costrai the matter desity fluctuatios more or less directly Ofte oe measures the clusterig of objects, whose positios are expected to reflect statistically the uderlyig clusterig of the matter desity field. I a Gaussia iitial desity field, objects are expected to exhibit biased clusterig with respect to the uderlyig DM power spectrum. Oe ofte sees: P gal (k) = b ( k)p k = b k P o k T k Jeas scale ~ horizo scale (the largest scale over which supportig pressure forces ca fuctio is the particle horizo) Horizo scale at matter-radiatio equality is imprited o power spectrum Low W m where bias parameter b is expected to be costat o large scales. 4. May 018 Frotiers of Cosmology - Mohr - Lecture May 018 Frotiers of Cosmology - Mohr - Lecture 38 Dampig effects I additio to havig growth retarded, small scale perturbatios ca actually be erased by dampig processes For collisioless matter, perturbatios are erased by simple free streamig- radom particle velocities lead to et loss from overdese regios ad et gai i uderdese regios. At sufficietly early times the dark matter particles were also relativistic ad could free stream Neutrio Impact o Power Spectrum Neutrios are low mass ad remai relativistic over much of the history of the Uiverse They free stream out of smaller scale perturbatios, leavig oly dark matter ad baryos behid Silk dampig is importat for baryos, because diffusio of photos out of overdese regios drags the leptos (ad their accompayig baryos) alog This leads to a reductio of power o small scales that icreases with the eutrio fractio 4. May 018 Frotiers of Cosmology - Mohr - Lecture May 018 Frotiers of Cosmology - Mohr - Lecture 40

11 Baryoic Features i the Power Spectrum Baryos are tightly coupled to the photos util recombiatio Thus, baryo perturbatios have oscillatory solutios (soud waves) Give the ~15% baryo fractio i the Uiverse, these features are observable APM Survey Results The agular correlatio fuctio aalysis was a critical step forward i testig structure formatio models It clearly demostrated icosistecy betwee the real uiverse ad the expectatios of the Stadard Cold Dark Matter (scdm, W M =1) model which was the theorists favorite at the time 4. May 018 Frotiers of Cosmology - Mohr - Lecture May 018 Frotiers of Cosmology - Mohr - Lecture 4 Observed Galaxy Power Spectrum Observatios of the galaxy power spectrum support adiabatic desity perturbatios. CMB aisotropy power spectrum is also cosistet with expectatios for adiabatic desity perturbatios SDSS Correlatio Fuctio The correlatio fuctio of the SDSS survey is show here, ad oe ca see a iterestig feature, which correspods to a baryoic feature at a scale of 150Mpc that is reflected i the distributio of galaxies See Eisestei et al 005 Plot from Tegmark page: gmark/sdss.html 4. May 018 Frotiers of Cosmology - Mohr - Lecture May 018 Frotiers of Cosmology - Mohr - Lecture 44

12 Spherical Harmoic Approach Similarly, oe ca use the agular power spectrum (Cooray et al 001) of these tracers Power Spectrum Costraits CMB Temperature Power Spectrum SPT + WMAP7 11 The agular power spectrum ivolves expadig the projected overdesity field i spherical harmoics δ θ = almylm θ () l,m () this plot adopts the D measure ad is plotted versus scale rather tha waveumber Oe the examies the equivalet of the D power spectrum, but i this case the treatmet accouts properly for the curved ature of the sky l lm C a Powerful set of tools available to work with maps- HEALPix 4. May 018 Frotiers of tegmark/sdss.html ad the optical depth are completely degeerate for the SPT badpowers, we impose a WMAP 7-based prior of = ± for the SPT-oly costraits. We preset the costraits o the CDM model from SPT data ad WMAP 7 data i colums two to four of Table 3. As show i Figure 5, the SPT badpowers costrai the CDM parameters approximately as well Cosmology as - Mohr - Lecture WMAP 7 aloe. The SPT ad WMAP 7 parameter costraits are cosistet for all parameters; s chages the most sigificatly amog the five free CDM parameters, movig by 1.5. The costrait o s also tightes the most from WMAP 7 to SPT. The SPT badpowers measure the agular soud horizo size extremely well by virtue of the sheer umber of acoustic peaks seve measured by the SPT badpowers. The SPT costrait o s is broader tha the costrait from WMAP 7 due to the fact that WMAP 7 probes a much greater dyamic rage of agular scales. Degeeracies with s degrade the SPT costraits o R, the baryo desity ad, to a lesser extet, the dark matter desity. Next, we preset the costraits o the CDM model from the combiatio of SPT ad WMAP 7 data. As previously metioed, we will refer to the joit SPT+WMAP 7 likelihood as the CMB likelihood. We the exted the discussio to iclude costraits from CMB data i combiatio with BAO ad/or H0 data. We preset the CMB costraits o the six CDM 45 parameters i the fourth colum of Table 3. Addig the full survey SPT badpowers tightes the costraits o b h, c h, ad by 33%, 9%, ad 31%, respectively, relative to WMAP 7. For compariso, the additio of the K11 badpowers to WMAP 7 led to improvemets of 5%, 14%, ad 10% respectively. The costrait o the scalar spectral idex is improved by 9%; these data costrai s < 1.0 at 3.9. The agular soud horizo scale at recombiatio, s, is set by the locatios of the acoustic peaks. Addig SPT data decreases the ucertaity o s by 63% relative to WMAP 7 aloe. The preferred values for b h ad c h do ot shift sigificatly from those iferred from WMAP 7 to those iferred from WMAP 7 + SPT ad therefore either Refereces Cosmological Physics, Joh Peacock, Cambridge Uiversity Press, 1999 Cosmological costraits from Galaxy Clusterig Will Percival (006) 4. May 018 Plot from Tegmark page: Fig. 4. The SPT badpowers (blue), WMAP 7 badpowers (orage), ad the lesed theory spectrum for the best-fit CDM cosmology show for CMB-oly (dashed lie), ad CMB+foregrouds (solid lie). As i Figure 3, the badpower errors show i this plot do ot iclude beam or calibratio ucertaities Combied CDM costraits ² Combied costraits o the power spectrum from a variety of tracers usig over a rage of scales 5x104 i physical scale ad dyamic rage i amplitude of 105 Frotiers of Cosmology - Mohr - Lecture May 018 Frotiers of Cosmology - Mohr - Lecture 46