Lecture VIII Dark Energy, Inflation and the Multiverse! are we alone, on the grandest scale? 12 A. B. Mantz et al. 0.0 0.0 0.5 0.5 − − ●● Clusters ●● CMB

1.0 1.0 ●

− − ● SNIa ●

w w ● BAO ●● All 1.5 1.5 − −

●● Clusters 2.0 2.0

− ●● CMB − ●● All 2.5 2.5 − − 0.6 0.7 0.8 0.9 1.0 1.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

σ8 Ωm

Figure 4. Constraints on constant-w dark energy models with minimal neutrino mass from our cluster data (with standard priors on h 2 2 and ⌦bh ) are compared with results from CMB (WMAP, ACT and SPT), supernova and BAO (also including priors on h and ⌦bh ) 2 data, and their combination. The priors on h and ⌦bh are not included in the combined constraints. Dark and light shading respectively indicate the 68.3 and 95.4 per cent confidence“Cosmic regions, harmony” accounting for systematic uncertainties.

over 0.5

Λ ● SNIa ΩΛ=0.6911+/-0.0062 0.8 ● free curvature (⌦k)andanevolvingequationofstate.In Ω ●● BAO H0=67.74+/-0.46all cases, the km/s/Mpc cluster data, and the combinations of cluster ● All 0.6 ● ageand of otherUniverse: leading data sets, remain consistent with spatial 13.799+/-0.021flatness and aGyr cosmological constant (see the right panel of

0.4 Figure 6 for models including free curvature). Comparing to M14,whouseidenticalfgas, CMB, supernova and BAO

0.2 data but not cluster counts, we generally find improvement in the constraints on w0, and less so for ⌦k and wa.Inthe

0.0 most general model we consider, the constraint on w0 shrinks +0.40 0.0 0.2 0.4 0.6 0.8 from 0.99 0.34 to 0.97 0.22 for the combination using ± +0.24 Ω WMAP CMB data (from 0.75 0.34 to 0.71 0.36 for the m ± combination using Planck+WP data). Figure 5. Constraints on ⇤CDM models (including curvature) with minimal neutrino mass from our cluster data (with stan- 2 dard priors on h and ⌦bh )arecomparedwithresultsfromCMB (WMAP, ACT and SPT), supernova and BAO (also including pri- 2 4.4 Constraints on Modifications of Gravity ors on h and ⌦bh )data,andtheircombination.Thepriorson 2 h and ⌦bh are not included in the combined constraints. Dark While dark energy (in the form of a cosmological constant) and light shading respectively indicate the 68.3 and 95.4 per cent has been a mainstay of the standard cosmological model confidence regions, accounting for systematic uncertainties. The since the discovery that the expansion of the Universe is dotted line denotes spatially flat models. accelerating, other explanations for acceleration are possi- ble. In particular, various modifications to GR in the large- scale/weak-field limit have been proposed (for recent reviews 3 prefers spatial flatness, with 10 ⌦k = 3 4and0 4 from see, e.g., Frieman et al. 2008; Clifton et al. 2012; Joyce et al. ± ± the combinations using WMAP and Planck+WP data, re- 2014). Being sensitive to the action of gravity in this regime, spectively. the growth of cosmic structure has the potential to distin- Turning to models with an evolving equation of state, guish between dark energy and modified gravity theories we first consider the simplest case without spatial curvature. that predict identical expansion histories. With this assumption, the cluster data alone are able to con- A simple and entirely phenomenological approach in- strain the w0 and wa parameters of the evolving dark energy volves modifying the growth rate of density perturbations model (see Equation 1), even when atr is free (marginalized at late times, when the growth is approximately scale-

c 2014 RAS, MNRAS 000, 1–22 The horizon problem

Two directions separated by more than how far light could travel before recombination (about 1 degree) could not “know” about each other. So why is the CMB temperature the same, to within dT/T~10-5? Like finding two civilisations on completely isolated islands in the Pacific and realising they speak the same language! a˙ 2 = 8⇡G⇢m a2 kc2 3 2 2 a˙ = H2 = 8⇡G(⇢m+⇢r +⇢⇤) kc a 3 a2

a˙ 2 2 8⇡G⇢⇤ a = H = 3 2 ⇢r = ur/c andThe⇢ flatness⇤ = ⇤ /problem8⇡G ⇢m,0 ⇢r,0 ⌦ = , ⌦ = , m ⇢cr,0 r ⇢cr,0 ⇢ ⌦We know= that⇤,0 todayand ⌦ = ⌦ + ⌦ + ⌦ is very close to 1. ⇤ ⇢cr,0 0 m r ⇤

Ω0 changes steeply as a function of z in the early Universe, so in order for Ω0 to be so close to 1 now, |1-Ω0|<10-15 at z=1010! How was such a “fine-tuning achieved”?

2 Inflation Particle physics so far well tested up to 100s of GeV. What if there is new physics at the scale of the Grand Unified Theories (GUT), ~1014 GeV (10-34s after the Big Bang), that causes the vacuum energy density to be much higher than today? Similarly to the dark-energy dominated Universe, we would get an exponential expansion at very early times!

from size of a nucleus to size of a galaxy: factor 1036

At early times, every part of the observable Universe was in causal contact. Explains horizon problem! Any “wrinkles” or curvature of space-time stretched out by inflation: solves flatness problem!

Bonus: solves “magnetic monopole” problem: many monopoles (magnetic “charge”) are predicted to be produced following Grand Unified Theories at high temperature, but none have been detected, possibly because they were so diluted during inflation.

It is assumed that then a phase transition took place where this vacuum energy density was transformed into normal matter and radiation, which stopped the “inflation”. dark matter dominated dark energy dominated (Lecture 4)

inflation dominated What is dark energy?

Yakov Zeldovich proposed in 1967 that dark energy is a quantum field phenomenon: Suppose physical vacuum is not really empty, but is filled with virtual particle-antiparticle pairs, which annihilate within Δt < ћ/mc2, and their Thefluctuations Worst give rise Scientific to a net energy density Prediction - a “ground” state Ever of the physical vacuumThe. Worst Scientific Prediction Ever • A “natural” Planck system of units expresses everything as To really estimate the• value A “natural” of this vacuum Planck energy system density, of units we expressesneed a quantum everything theory as of combinationgravity, which we don’tof fundamental havecombination yet. A “natural” of fundamental physical Planck system physical constants; of units constants; expresses the the everything Planck densityas a combination is:! of fundamentaldensity is: physical! constants; the Planck density is: ρ = c 5 / (ћ G 2) = 5.15 !10 +93 g cm-3! 5 Planck 2 +93 -3 ρPlanck =• Thec observed/ (ћ G value ) = is: !5.15 !10 g cm ! −30 -3 • The observedObserved value value is::! ρvac = Ωvac ρcrit ≈ 6.5 ! 10 g cm ! Ooops! Off by 123 orders of magnitude …! ρ = Ω ρ ≈ 6.5 ! 10 −30 g cm-3! Off by vac123• ordersThisvac is of modestly magnitude:crit called worst“the fine-tuning prediction problem” in history! (because it 123 Ooops! The other Offrequires “natural” by a123 cancellation value orders is zero, to 1 whichpart of in magnitude 10is also)! wrong. …! • The other “natural” value is zero! • This is modestly• So,called lacking “the a proper fine-tuning theory, physicists problem” just declared (because the it requires a cancellationcosmological to 1 constant part into be10 zero,123 )and! went on…! • The other “natural” value is zero! • So, lacking a proper theory, physicists just declared the cosmological constant to be zero, and went on…! What are the physical origins of the cosmological constant?

At the moment, completely unknown — one of the biggest mysteries in modern astrophysics! Even more mysteries: Why is ΩΛ non-zero but so much smaller than the “natural” Planck value? Why is ΩΛ ~ Ωm today, when they could be orders of magnitude off? Many of the proposed models are based on one of the following: – Decay of some scalar field (e.g. quintessence, proposing a fifth fundamental force) – Modified theories of gravity – Models connecting DM and DE – Landscape or multiverse models that postulate the existence of ~10500 separate universes, with different (random) values of the physical constants, Λ included and so on!

One measurement that might help eliminate some possibilities is

a possible time evolution of ΩΛ a˙ 2 = Is8⇡ theG⇢m cosmologicala2 kc2 constant a constant? 3 2 2 a˙ = H2 = 8⇡G(⇢m+⇢r +⇢⇤) kc a 3 a2 1 8 7.9? 8.1? a˙ 2 2 8⇡G⇢⇤ a = H = 3 2 ⇢r = ur/c and ⇢⇤ = ⇤/8⇡G dark energy ⇢m,0 ⇢r,0 ⌦ = , ⌦ = , m ⇢cr,0 r ⇢cr,0 ⇢⇤,0 ⌦Suppose= ρΛ=ρandΛ,0a-3(1+w)⌦ : =if w=-1,⌦ +then⌦ ρΛ+ is truly⌦ constant (does not ⇤ ⇢cr,0 0 m r ⇤ change with a). Can we see any deviations2 of w from -1? 2 2 ⌦r ⌦m kc H (t)=H0 4 + 3 + 2 2 + ⌦⇤ a (t) a (t) H0 a (t) 2 2 h ⌦r ⌦m 1 ⌦0 i H (t)=H0 a4(t) + a3(t) + a2(t) + ⌦⇤ 3H2 0 h i 3(1+w) ⇢c,0 = 8⇡G ΩΛ/a Or, suppose w=-1 at the present time, but what if it changes with time such that for example w(a)=w0+wa(1-a)?

2 14 A. B. Mantz et al. 2 ●● Clusters

Current constraints on 5 ●● CMB 1 dark matter equation ●● SNIa ●● BAO 0 of state ●● All 1 0 ΛCDM − a a w w 12 A. B. Mantz et al. 2 − ρΛ=ρΛ,0a-3(1+w) 3 − 5 0.0 0.0 − ● 4 ● at = 0.5 − ●● at free 0.5 0.5 5

− − ΛCDM ● −

● Clusters 10 ●● CMB −

1.0 1.0 ● −4 −2 0 2 −1.5 −1.0 −0.5 0.0

− − SNIa ● Planck Collaboration: Cosmological parameters ● w w w w ● BAO w(a)=w0+wa(1-a) 0 0 ● This is consistent with the preference for● aAll higher lensing am- 2 1.5 1.5 plitude discussed in Sect. 5.1.2, improving the fit in the w

− − Planck TT+lowP+ext 2 region, where the lensing smoothing amplitude becomes slightlydata (with standard priors on h and ⌦bhPlanck)arecomparedwithresultsfromCMB(WMAP,ACTandSPT),supernovaandBAO(alsoTT+lowP+WL 2 2 larger. However, the lower limit in Eq. (51) is largely determinedincluding priors1 on h and ⌦bh )data,andtheircombination.ThepriorsonPlanck TT+lowP+WL+H0 h and ⌦bh are not included in the combined constraints. ● 1 1 ● Clusters by the (arbitrary) prior H0 < 100 km s Mpc , chosen forDark the and light shading respectively indicate the 68.3 and 95.4 per cent confidence regions, accounting for systematic uncertainties. The 2.0 2.0

− ●● CMB − Hubble parameter. Much of the posterior volume in the phan-cross indicates the ⇤CDM model (w0 = 1, wa =0).Right:Constraintsonevolving-w models with global curvature as a free parameter tom region is associated with extreme values for cosmological ● All from the combination0 of cluster, CMB, supernova and BAO data. For the model with atr free, this parameter is marginalized over the ● parameters, which are excluded by other astrophysical data. The range 0.5a

⇤ w 2.5 2.5 mild tension with base CDM disappears as we add more data − − that break the geometrical degeneracy. Adding Planck lensing 1 0.6 0.7 0.8 0.9 1.0 1.1 0.0 and0.1 BAO, JLA0.2 and H0.30 (“ext”)0.4 gives the0.5 95 %0.6 constraints0.7 +0.091 Table 3. Marginalized best-fitting values and 68.3 per cent tivariate Gaussian, encoding measurements of f8(z)and σ8 w = 1.023 0.096 PlanckΩmTT+lowP+ext, (52a) 2 F (z)=(1+z)D(z)H(z)/c at several redshifts, assuming +0.085 maximum-likelihood confidence intervals for the growth index (), w = 1.006 0.091 Planck TT+lowP+lensing+ext, (52b) 8,andw from clusters (Cl), the CMB and galaxy survey data zero neutrino mass; here D is the angular diameter distance, Figure 4. Constraints on constant-w dark energy models with minimal neutrino mass= from. our+0.075 cluster data (with, , standard+ + priors+ on. h 2 w 1 019 0.080 Planck TT TE EE lowP lensing ext 2(gal). Here determines the late-time growth of cosmic struc- and c is the speed of light. For consistency, we fix m⌫ =0 and ⌦bh ) are compared with results from CMB (WMAP, ACT and SPT), supernova and BAO (also including priors on h and ⌦bh ) 3 Mantz et al. 2015, (52c)ture, and w should be interpreted purely as a modification to the data, and their combination. The priors on h and ⌦ h2 are not included in the combined constraints. Dark and light shading respectively 2 1 0 1 in this section for all data sets, rather than using the base- b Planck Collaboration 2015 ⇤CDM expansion model (but not directly to the growth). Sub- P indicate the 68.3 and 95.4 per cent confidence regions, accounting for systematic uncertainties.The addition of Planck lensing, or using the full Planck tem- w0 line value of 0.056 eV employed elsewhere in this paper. Due perature+polarization likelihood together with the BAO, JLA,scripts ‘WM’ and ‘Pl’ denote the use of WMAP or Planck+WP a to the approximate nature of the galaxy survey likelihood and H0 data does not substantially improve the constraintdata of in combination with ACT and SPT. Note: the combina- Fig. 28. Marginalized posterior distributions for (w0, wa) for var- used here, compared with the analysis of cluster and CMB Eq. (52a). All of these data set combinations are compatible withtions withious data galaxy combinations. survey data We should show Planck be treatedTT+ withlowP caution in combi- due ⇤ = data, we urge caution in interpreting the results that com- over 0.5

Ω tion of Planck+CFHTLenS± pulls the contours± into the phantom over the standard set of free parameters of the flat ⇤CDM ●● BAO the SNe data in the JLA sample. One consequence of this is theCl+CMBPl+gal 0.67 0.06 0.799 0.015 1 all cases, the cluster data, and the combinations of cluster domain and is discrepant with± base ⇤CDM at± about the 2 level. ● All tightening of the errors in Eqs. (52a)–(52c) around the ⇤CDM model. In the case of CMB or galaxy survey data alone, there 0.6 ● and other leading data sets, remain consistent with spatialCl+ 0.39 0.24 0.850 0.055 0.90 0.19 value w = 1 when we combine the JLA sample with Planck. The Planck CFHTLenS data± also favour a± high value of H0. If± are strong but complementary degeneracies (as discussed Cl+CMBwe add the (relatively0. weak)52 0H.14prior 0. of817 Eq. (030.040), the contours0.94 0.13 flatness and aIf cosmologicalw di↵ers from constant1, it is likely (see to the change right with panel time. of We WM ± 0 ± ± by Rapetti et al. 2013), whereas the cluster data (with Cl+CMB(shown in+gal cyan) in Fig. 0.6028 shift0.08 towards 0.792w =0.0201. It therefore0.91 0.08

0.4 consider here the case of a Taylor expansion of w at first order in WM Figure 6 for models including free curvature). Comparing ± ± ± standard priors) constrain the entire model; the marginal- the scale factor, parameterized by Cl+CMBseemsPl unlikely that the0.57 tension0.14 between 0.828Planck0.and040 CFHTLenS1.01 0.13 to M14,whouseidenticalfgas, CMB, supernova and BAO can be resolved by allowing± a time-variable± equation of state for± ized constraints from clusters are =0.48 0.19 and Cl+CMBPl+gal 0.63 0.07 0.799 0.015 0.96 0.07 0.2 w = w0 + (1 a)wa. (53) ± ± ± ± data but not cluster counts, we generally find improvement dark energy. 8 =0.83 0.05. More complex models of dynamical dark energy are discussed A much more extensive investigation of models of dark ± in the constraints on w0, and less so for ⌦k and wa.Inthe All three data sets shown are individually consistent

0.0 in Planck Collaboration XIV (2016). Figure 27 shows the 2D energy and also models of modified gravity can be found in most general model we consider, the constraint on w0 shrinks with =0.55. Their combination has a marginal (< 2) marginalized posterior+0 distribution.40 for w0 and wa for the com-dominatedPlanck Collaboration regime, where XIVf(20161independentof). The main conclusions) of onward that 0.0 0.2 0.4 0.6 0.8 from 0.99bination0.34Planck to +0BAO.97 +0JLA..22 for The the JLA combination SNe data are again using cru- analysis are as follows: ! preference for higher values of (Table 3), though this ± +0.24 to be consistent with the growth given by Equation 14.This Ω WMAP CMBcial data in breaking (from the geometrical0.75 0.34 degeneracy to 0.71 at low0.36 redshiftfor the and should be viewed with caution in light of the caveats men- m ± modified power spectrum is then integrated when evaluating combinationwith using thesePlanck data we+WP find no data). evidence for a departure from the an investigation of more general time-variations of the equa- tioned above (see also Beutler et al. 2014b). The combina- base ⇤CDM cosmology. The points in Fig. 27 show samplesthe cluster• mass function (Equations 2–3). For details of the Figure 5. Constraints on ⇤CDM models (including curvature) tion of state shows a high degree of consistency with w = 1; tion of clusters and the CMB (without galaxy survey data) from these chains colour-coded by the value of H0. From thesecalculation of the ISW e↵ect in this model, see Appendix C; 1 1 a study of several dark energy and modified gravity models with minimal neutrino mass from our cluster data (with stan- MCMC chains, we find H0 = (68.2 1.1) km s Mpc . Much • is fully consistent with GR. 2 ± as in earliereither sections,finds compatibility we use with CMB base data⇤CDM, from or mild ACT, tensions, SPT, dard priors on h and ⌦bh )arecomparedwithresultsfromCMB higher values of H would favour the phantom regime, w < 1. 0 and eitherwhichPlanck are driven+WP mainly or WMAP. by external The data galaxy sets. survey data In the right panel of Figure 7, we present constraints (WMAP, ACT and SPT), supernova and BAO (also including pri- 4.4 ConstraintsAs pointed on out Modifications in Sects. 5.5.2 and of5.6 Gravitythe CFHTLenS weak 2 lensing data are in tension with the Planck base ⇤CDM param-include results from 6dF (Beutler et al. 2012), SDSS (Reid on models when additional freedom is introduced into the ors on h and ⌦bh )data,andtheircombination.Thepriorson 2 eters. Examples of this tension can be seen in investigationset of al.6.4.2012 Neutrino) and physics the WiggleZ and constraints Dark Energy on relativistic Survey (Blake model for the cosmic expansion in the form of the w pa- h and ⌦bh are not included in the combined constraints. Dark While dark energy (in the form of a cosmological constant) dark energy and modified gravity, since some of these modelset al. 2011components). Their likelihood is approximated by a mul- rameter. In this model, w should not be interpreted as the and light shading respectively indicate the 68.3 and 95.4 per cent has been acan mainstay modify the of growth the rate standard of fluctuations cosmological from the base model⇤CDM confidence regions, accounting for systematic uncertainties. The since the discoverypredictions. that This tension the expansion can be seen even of the in the Universe simple model is In the following subsections, we update Planck constraints on c 2014 RAS, MNRAS 000, 1–22 dotted line denotes spatially flat models. accelerating,of other Eq. (53). explanations The green regions for in acceleration Fig. 28 show 68 are % and possi- 95 % the mass of standard (active) neutrinos, additional relativistic de- contours in the w0–wa plane for Planck TT+lowP combined with grees of freedom, models with a combination of the two, and ble. In particular,the CFHTLenS various H13 modifications data. In this example, to GR we have in the applied large- ultra- models with massive sterile neutrinos. In each subsection we scale/weak-fieldconservative limit cuts, have excluding been proposed⇠ entirely (for and excluding recent reviews measure- emphasize the Planck-only constraint, and the implications of 3 ments with ✓ < 17 in ⇠ for all tomographic redshift bins. As the Planck result for late-time cosmological parameters mea- prefers spatial flatness, with 10 ⌦k = 3 4and0 4 from see, e.g., Frieman et al. 20080 ; Clifton+ et al. 2012; Joyce et al. ± ± discussed in Planck Collaboration XIV (2016), with these cuts sured from other observations. We then give a brief discussion of the combinations using WMAP and Planck+WP data, re- 2014). Beingthe sensitive CFHTLenS to data the are action insensitive of gravity to modelling in this the regime, nonlinear tensions between Planck and some discordant external data, and spectively. the growthevolution of cosmic of the structure power spectrum, has the but potential this reduction to in distin- sensitiv- assess whether any of these model extensions can help to resolve Turning to models with an evolving equation of state, guish betweenity comes dark at theenergy expense and of reducing modified the statistical gravity power theories of the them. Finally we provide constraints on neutrino interactions. we first consider the simplest case without spatial curvature. that predict identical expansion histories. With this assumption, the cluster data alone are able to con- A simple40 and entirely phenomenological approach in- strain the w0 and wa parameters of the evolving dark energy volves modifying the growth rate of density perturbations model (see Equation 1), even when atr is free (marginalized at late times, when the growth is approximately scale-

c 2014 RAS, MNRAS 000, 1–22 Weighing the Giants IV: Cosmology and Neutrino Mass 13

Table 2. Marginalized (one-dimensional) best-fitting values and 68.3 per cent maximum-likelihood confidence intervals for the pa- rameters of various dark energy models, including systematic uncertainties. The parametrization of the equation of state is defined in Section 3.1.The“Clusters”dataincorporatesX-raysurveydata,X-rayfollow-upobservations(providingmassproxiesingeneraland fgas measurements for relaxed clusters), and weak lensing data (WtG). The “CombWM”combinationofdatareferstotheunionofour cluster data set with CMB power spectra from WMAP (Hinshaw et al. 2013), ACT (Das et al. 2014)andSPT(Keisler et al. 2011; Reichardt et al. 2012; Story et al. 2013), the Union 2.1 compilation of type Ia supernovae (Suzuki et al. 2012), and baryon acoustic oscillation measurements at z =0.106 (Beutler et al. 2011), z =0.35 (Padmanabhan et al. 2012)andz =0.57 (Anderson et al. 2014). “CombPl”isidentical,withtheexceptionthat1-yearPlanck data (plus WMAP polarization; Planck Collaboration 2013c)areusedin 2 place of the complete 9-year WMAP data. The clusters-only constraints incorporate standard priors on h and ⌦bh (Section 2; Riess et al. 2011; Cooke et al. 2014). Parameters listed with no error bars for a given model are fixed. Parameters with no value listed are not relevant, given the other parameters that are fixed in that model. For the models in which wa is a free parameter (bottom section of table), there is no sensitivity to the transition time parameterized by atr;therefore,atr is either fixed (to 0.5) or is marginalized over the range 0.5 to 0.95 (indicated by the “—” symbol in the atr column). The last column indicates in which figure, if any, the corresponding results are displayed.

3 Data 8 ⌦m ⌦DE 10 ⌦k w0 wa wet atr Fig.

Clusters 0.830 0.035 0.259 0.030 0 10 1 ± ± Clusters 0.830 0.035 0.261 0.032 0.728 0.115 8 110 10 5 ± ± ± ± Comb 0.814 0.019 0.294 0.010 0.709 0.011 3 4 10 5 WM ± ± ± ± Comb 0.823 0.013 0.302 0.009 0.698 0.009 0 4 10 B1 Pl ± ± ± ± Clusters 0.831 0.036 0.261 0.031 0 0.98 0.15 0 4 ± ± ± Comb 0.819 0.026 0.295 0.013 0 0.99 0.06 0 4 WM ± ± ± Comb 0.833 0.021 0.297 0.013 0 1.03 0.06 0 B1 Pl ± ± ± Comb 0.818 0.023 0.289 0.014 0.715 0.016 5 5 1.03 0.07 0 WM ± ± ± ± ± Comb 0.836 0.021 0.292 0.014 0.713 0.015 4 4 1.08 0.07 0 Pl ± ± ± ± ± +0.32 +1.9 +1.6 Clusters 0.829 0.036 0.261 0.026 0 0.69 0.36 1.6 1.3 2.3 1.0 0.5 6a ± ± +0.13 +0.4 +0.3 CombWM 0.816 0.027 0.292 0.015 0 1.04 0.18 0.3 0.6 0.8 0.4 0.5 6a ± ± +0.15 +0.6 +0.4 CombPl 0.835 0.021 0.298 0.015 0 0.96 0.18 0.3 0.5 1.2 0.4 0.5 B1 ± ± +0.62 +1.5 +0.8 Clusters 0.827 0.036 0.262 0.023 0 0.71 0.42 1.0 1.4 1.4 1.1 — ± ± +0.23 +0.5 +0.2 CombWM 0.818 0.025 0.291 0.015 0 1.09 0.22 0.2 0.5 0.9 0.2 — ± ± +0.24 +0.5 +0.2 CombPl 0.834 0.021 0.300 0.015 0 0.97 0.20 0.2 0.5 1.1 0.3 — ± ± +0.24 +1.0 +0.8 CombWM 0.822 0.026 0.294 0.015 0.713 0.016 8 6 0.93 0.20 0.4 1.1 1.3 0.9 0.5 6b ± ± ± ± +0.26 +0.9 +0.7 CombPl 0.840 0.022 0.302 0.015 0.705 0.016 8 5 0.87 0.20 0.8 1.4 1.6 1.1 0.5 B1 ± ± ± ± +0.40 +0.6 +0.5 CombWM 0.822 0.025 0.295 0.016 0.712 0.016 7 7 0.97 0.22 0.1 1.2 1.1 0.7 — 6b ± ± ± ± +0.24 +1.1 +0.3 CombPl 0.838 0.021 0.304 0.016 0.703 0.016 7 5 0.71 0.36 1.1 0.7 1.3 0.8 — B1 ± ± ± ±

Current constraintsindependent. on deviations Wefrom adopt general the relativity simple parametrization in terms from other data sets have been obtained by, e.g., Nesseris What if we modify the growthof therate growthof density index, perturbations (e.g. atLinder late times2005, ), & Perivolaropoulos (2008), di Porto & Amendola (2008), naively parametrised as: Samushia et al. (2013, 2014)andBeutler et al. (2014a). d ln f(a)= = ⌦m(a) , (14) We follow Rapetti et al. (2013), investigating the con- Weighing the Giantsd IV:ln a Cosmology and Neutrino Mass 15 δ is the density contrast straints on from our cluster data, the integrated Sachs- 26 2.5 compared to the mean where is the linear density contrast in synchronous gauge Wolfe (ISW) e↵ect on the CMB, and measurements of ●● Clusters (at any1.2 scale), and where =0.55 approximately corre- density. From ●GR,CMB we expect redshift-space distortions (RSD) and the Alcock-Paczynski 2.0 ● γ=0.55. Do we●● seeGalaxies any sponds to1.0 GR for a wide range of expansion histories com- (AP) e↵ect from galaxy survey data. In practice, we use

1.5 deviations of the law of gravity patible with0.8 current data (Polarski & Gannouji 2008). Note camb to calculate and tabulate P (k, z)assumingGR,then on4 large scales? VIKHLININ ET AL. that constraints on the growth indexGR serve only as a use- 0.6 1.0 γ γ modify these values from z = 30 (well into the matter- ΩM =0GR.25, ΩΛ =0.75, h =0.72ful consistencyΩM =0 check.25, ΩΛ =0, h =0 of.72 GR, rather than directly testing 10−5 10−5 0.4 0.5 GR against alternative models of gravity. Constraints on −6 −6 3 10 3 10 − − from earlier0.2 versions of our cluster analysis (in conjunction 26 0.0 Mpc Mpc Cosmic growth also leaves an imprint at high multipoles 3 3 ● − − −7 −7 ● Clusters h h 10 10 with contemporaneous● cosmological data) are presented by through CMB lensing, but currently the CMB constraints on , ,

) Cl+CMB ) ● 0.5 ΛCDM >M − >M ( ( −8 −8 0.2

N primarily come from the ISW e↵ect. N 10 Rapetti10 et al. (2009, 2010, 2013). Independent constraints −

0.4 0.6 0.8 −9 1.0 1.2 1.4 1.6 −9 −1.6 −1.2 −0.8 −0.4 10 z =0.025 − 0.25 10 z =0.025 − 0.25 σz8=0.55 − 0.90 c 2014 RAS,z =0.55 − 0.90 MNRAS 000w , 1–22 14 15 1014 Mantz et1015 al. 2015 Figure 7. Constraints on models where10 the growth index of10 cosmic structure formation, ,isafreeparameter.Darkandlightshading−1 M h−1 M M500, h M⊙ respectively indicate the 68.3 and 95.4 per cent confidence500, ⊙ regions, accounting for systematic uncertainties. Left: Constraints from clusters, F￿￿. 2.— Illustration of sensitivity of the cluster mass function to the cosmological model. In the le￿ panel, we show the measured mass function and predicted the CMB, and galaxy surveymodels data (with only individually, the overall normalization marginalizing at z 0 adjusted) computed over for a the cosmology standard which is close flat to our best-⇤CDM￿t model. ￿ parametrizatione low-z mass function is reproduced of the cosmic expansion history. Note that the treatmentfrom Fig. 1, of which the for the galaxy high-z cluster survey we show data only the most uses distant a multivariate subsample (z 0.55) to Gaussian better illustrate the approximation e￿ects. In the right panel, to both constraints the data and the from RSD and the models are computed for a cosmology with ΩΛ = 0. Both the model and the data at high redshi￿sarechangedrelativetotheΩΛ 0.75 case. ￿e measured mass AP e↵ect (see also Rapettifunction et al. is changed2013 because). GR it is correspondsderived for a di￿erent distance-redshi approximately￿ relation. ￿ toe> model =0 is changed.55 because (dashed the predicted line). growth Right: of structure Constraints and overdensity from clusters and thresholds corresponding to ∆crit 500 are di￿erent.= When the overall model normalization is adjusted to the low-z mass function,= the predicted number density the combination of clustersof andz 0.55 the clusters CMB is in strong for disagreement models with where the data, andw thereforeis allowed this combination to be of Ω freeM and Ω inΛ can the be rejected. parametrization of the expansion history (this = parameter does not directly a↵> ect the growth history in this model). Here the horizontal and vertical dashed lines respectively correspond of interest in our study; at this level, the theoretical uncertain- § 4.2.8 of Dunkley et al. 2008) and so we experiment also with to the standard models forties the in the growth mass function of cosmic do not contribute structure signi￿ (GR)cantly to and the theneutrino expansion masses outside of the the Universe Ly-α forest bounds (⇤CDM). (§ 8.5). In these figures, ‘CMB’ refers to the combination ofsystematic ACT, error SPT budget. and Although WMAP the formula data; has see been Appendix cali- B for the corresponding figures using Planck+WP instead of brated using dissipationless N-body simulations (i.e. without 4. FITTING PROCEDURE WMAP data. e￿ects of baryons), the expected e￿ect of the internal redistri- We obtain parameter constraints using the likelihood func- bution of mass during baryon dissipation on halo mass func- tion computed on a full grid of cosmological parameters a￿ect- tion are expected to be 5% (Rudd et al. 2008) for a realistic ing cluster observables (and also those for external datasets). fraction of baryons that condenses to form galaxies. ￿e relevant parameters for the cluster data are those that a￿ect dark energy equation of state,Similarly but to Jenkins simply et< al. (2001) as a and phenomeno- Warren et al. (2006), the Importantly,the distance-redshi the￿ relation, cluster as well signal as the growth is influenced and power by the entire se- Tinker et al. formulas for the halo mass function are presented spectrum of linear density perturbations: ΩM, ΩΛ, w (dark en- logical departure from theas a cosmic function of expansion variance of the density model￿eld given on a mass by scale M. riesergy of equationn-point of state parameter),correlationσ8 (linear functions amplitude of ( den-Lo Verde et al. 2008; ⇤CDM, in the same way￿ thate variance, parametrizes in turn, depends on the departures linear power spectrum of of Shanderasity perturbations et al.at the2013a 8 h 1 Mpc), scale and at z therefore0), h,tiltofthe has the potential to the cosmological model, P k , which we calculate as a product − n primordial ￿uctuations power spectrum, and potentially, the the growth history fromof that the initial given power by law GR. spectrum, (Ink , particular, and the transfer func- distinguishnon-zero rest mass competing of light neutrinos. models￿is is= computationally of inflation that have identi- tion for the given mixture( of) CDM and baryons, computed demanding and we describe our approach below. dark energy perturbationsusing associated the analytic approximations with values of Eisenstein of w &dif- Hu (1999). cal￿ bispectrae computation but of the a likelihood di↵erent function scaling for a single of com- higher-order moments ferent from 1 are not included￿is analytic approximation in the growth is accurate equations, to better than 2% for (e.g.binationBarnaby of parameters & is Shandera relatively straightforward.2012). Our pro- a wide range of cosmologies, including cosmologies with non- cedure (described in Paper II) uses the full information con- which instead depend onnegligible through neutrino Equation contributions to14 the.) total The matter fig- density. tained in the dataset, without any binning in mass or redshi￿, Our default analysis assumes that neutrinos have a negligi- takesShandera into account the et scatter al. (2013b in the Mtot) presentvs. proxy relations constraints on two such ure shows constraints frombly small clusters mass. ￿ alone,e only component and the of our combina- analysis that could and measurement errors, and so on. We should note, how- be a￿ected by this assumption is when we contrast the low- inflation models, referred to as hierarchical-type (single- tion of cluster and CMB data. Here again, the clusters and ever, that since the measurement of the Mgas and YX proxies redshi￿ value of σ8 derived from clusters with the CMB power fielddepends inflation) on the assumed and distance feeder-type to the cluster, the (including mass func- interactions with clusters+CMB data arespectrum fully consistent normalization. with￿is comparison the standard uses evolution of tions must be re-derived for each new combination of the cos- purely CDM+baryons power spectra. ￿e presence of light aspectatorfield),basedonthemological parameters that a￿ect the distance-redshiM10a￿ relation,b data set. In this w = 1, =0.55 model,neutrinos although a￿ects the the power full spectrum combination, at cluster scales; in- in terms work,— ΩM, thew, ΩΛ free, etc. Variations parameter of h lead describing to trivial rescalings the of overall level of non- cluding the galaxy surveyof data,σ8,thee exhibits￿ect is roughly mild proportional (< 2 to)tension the total neutrino the mass function and do not require re-computing the mass density, and is 20% for mν 0.5 eV (we calculate the ef- Gaussianityestimates. Computation is the of dimensionless the survey volume uses third a model moment for of the density fect of neutrinos using the transfer function model of Eisen- (Table 3). the evolving LX Mtot relation (see § 5 in Paper II), which is 1 stein & Hu 1999).≈ Stringent∑ upper= limits on the neutrino mass perturbationmeasured internally field, from the smoothed data and thus also on depends scales on of 8h Mpc, 3; were reported from comparison of the WMAP and Ly-α forest M thethe assumedtwo modelsd z− function. above￿erefore, di↵ weer re in￿tthe theLX scalingMtot of higher-order data, mν 0.17 eV at 95% CL (Seljak et al. 2006). If neutrino relation for each new cosmology and recompute V M . Sen- masses are indeed this low, they would have no e￿ect on our 4.5 Constraints on Non-Gaussianity momentssitivity of the relative derived( ) mass to function3, to and the background in the− cos-form of the modified, analysis.∑ However,< possible issues with modeling of the Ly-α mology is illustrated in Fig. 2. ￿Me entire procedure,( although) data have been noted in the literature (see, e.g., discussion in non-Gaussianequivalent to full reanalysis mass offunction. the Chandra Inand particular,ROSAT data, the feeder scaling In the standard cosmological model, the primordial density generates greater non-Gaussianity overall for a given value perturbations sourced by inflation are assumed to be Gaus- of 3 than the hierarchical scaling. sian, in which case their statistical properties are completely M described by the power spectrum (i.e. two-point correla- More recently, Adhikari et al. (2014) have performed N- tion function). However, many viable inflation models pro- body simulations of structure formation from non-Gaussian duce non-Gaussianity, which results in non-vanishing higher- initial conditions. Their results for non-Gaussian mass func- order correlations (see, e.g., Bartolo et al. 2004). CMB and tions broadly vindicate the analytic approach of Shandera galaxy survey studies of non-Gaussianity typically focus on et al. (2013b), but motivate several refinements of the model, constraining the amplitude of the bispectrum (three-point detailed in Adhikari et al. (2014), which we adopt here. function), parametrized by fNL, for a given “triangle” tem- We do not recapitulate these refinements here, but note plate configuration of momentum vectors (e.g. Bennett et al. that their net e↵ect is to reduce the modification to the 2013; Planck Collaboration 2013e). mass function for a given value of 3 compared with the M For clusters, non-Gaussianity manifests itself in an en- Shandera et al. (2013b) model, for both hierarchical- and hancement or suppression of the mass function at the high- feeder-type scalings. Consequently, our constraints on non- est masses, and respectively a corresponding suppression or Gaussianity are weaker than those reported by Shandera enhancement at low masses, relative to the Gaussian case. et al. (2013b), despite our addition of lensing data to the c 2014 RAS, MNRAS 000, 1–22 The fate of the Universe: Are there any observational ways to constrain inflation?

The explosive expansion of space during inflation would have created ripples in the fabric of space. These gravity waves should leave a signature in the polarization of the CMB.

The amplitude of the gravity wave is proportional to the expansion rate during inflation, which in turn is proportional to the inflation energy scale squared.

+ The simplest models of inflation predict a power-law primordial density fluctuation spectrum, but more complex models suggest a deviation from a power-law spectrum (“running spectral index”) which can be measured. CMB polarization

Isotropic incident radiation — scattered light is unpolarised. If the incident radiation field has a quadrupolar variation in intensity or temperature (with intensity peaks at 90 degree separations), the result is a linear polarization of the scattered radiation.

Scalar perturbations: The fluid velocity from hot to colder regions cause blueshift of the photons, resulting in quadrupole anisotropy. Tensor perturbations: Gravity waves stretch and squeeze space in orthogonal directions, also stretching the wavelength of radiation, therefore creating a quadrupole variation. (Vector perturbations should be damped by inflation) The polarization pattern in the sky can be decomposed into 2 components: Curl-free component, called "E-mode" (electric-field like) or "gradient-mode", mirror- symmetric Grad-free component, called "B-mode" (magnetic-field like) or “curl-mode", not mirror- symmetric The E-mode may be due to both the scalar and tensor perturbations, but the B-mode is due to only vector or tensor perturbations because of their handedness (symmetry). B- mode polarisation can also be introduced by gravitational lensing.

E-mode polarisation pattern from stacking more than 11,000 cold (blue) and 10,000 hot (red) spots in the CMB with Planck. Current measurements of B-mode polarization

Currently, only upper limits on B-mode polarisation from primordial gravitational waves during inflation. Any detection here would tell us the inflation energy scale Current upper limits on B-mode polarisation from primordial gravitational waves and the primordial spectral index already starting to rule out some theoretical models of inflation!

simple single-field inflationary models predicted ns significantly different from 1, which is observed! More cosmic fine-tuning (1) the ratio of the strength of the electrical force compared to gravity: 1012; if gravity were just a few orders of magnitude stronger, no creatures could grow larger than insects, and there would be no time for biological evolution. (2) if the strong interaction were even a bit stronger than now, nuclear fusion would have converted all hydrogen in the early universe to helium — no water, therefore no life! If the strong interaction were even a bit weaker, H would not fuse to He at all — no stars! (3) if the total matter density were too high, the Universe would have re-collapsed very early on; had it been too low, no galaxies or stars would have formed. (4) had the cosmological constant been too large, the Universe would have been ripped apart much sooner — no time for life to develop! (5) the level of perturbations in the initial density field. If too small, no structure can form before dark energy takes over. If too large, the Universe would be full of black holes instead. (6) the number of (macroscopic) spatial dimensions. Life as we know it would not exist in 2 or 4D! (inverse-square law in 3D becomes inverse-cube in 4D, orbits of planets become unstable!) recommended reading: “Just Six Numbers: The Deep Forces that Shape the Universe” by Martin Rees Multiverse Level I: if inflation happened, there are almost certainly regions far away in space, beyond the sphere from which light has had time to reach us during the 13.7 billion years since our Big Bang. The apparent laws of physics are the same, but history played out differently because things started out differently (e.g. different curvature, matter density, level of initial fluctuations etc.) Level II: if you add in string theory with a landscape of possible solutions, these very distant parts of the universe could conceivably have different physics laws (different number of spatial dimensions, different physical constants, different elementary particles, etc.) Would explain “cosmic fine tuning” Level III: parallel worlds elsewhere in the so-called Hilbert space where quantum reality plays out Level IV: totally disconnected realities governed by different mathematical equations.

http://space.mit.edu/home/tegmark/crazy.html Suggested essay topic

What do the TE and EE power spectra of the CMB look like from Planck measurements, and how were they used to improve the cosmological constraints obtained from using only the temperature fluctuations?