Dark Energy Theory Overview
Ed Copeland -- Nottingham University 1. Issues with pure Lambda 2. Models of Dark Energy 3. Modified Gravity approaches 4. Testing for and parameterising Dark Energy
Dark Side of the Universe - Bergen - July 26th 2016
1 M. Betoule et al.: Joint cosmological analysis of the SNLS and SDSS SNe Ia.
46 The Universe is sample coh low-z 0.12 HST accelerating and
C 44
SDSS-II 0.11 yet we still really
SNLS 0.08 have little idea 1 42 HST 0.11 X what is causing
↵ SNLS Table 9. Values of coh used in the cosmological fits. Those val- + 40 this acceleration. ues correspond to the weighted mean per survey of the values ) G
( SDSS shown in Figure 7, except for HST sample for which we use the 38 Is it a average value of all samples. They do not depend on a specific M cosmological B choice of cosmological model (see the discussion in §5.5). ? 36
m constant, an Low-z = 34 evolving scalar µ 0.2 field, evidence of modifications of 0.4 . General
CDM 0 2 ⇤ 0.0 0.15 µ Relativity on .
0 2 large scales or µ 0.4 2 1 0 something yet to 10 10 10 coh 0.1 z be dreamt up ? σ Betoule et al 2014 2 Fig. 8. Top: Hubble diagram of the combined sample. The dis- tance modulus redshift relation of the best-fit ⇤CDM cosmol- 0.05 1 1 ogy for a fixed H0 = 70 km s Mpc is shown as the black line. Bottom: Residuals from the best-fit ⇤CDM cosmology as a function of redshift. The weighted average of the residuals in 0 0 0.5 1 logarithmic redshift bins of width z/z 0.24 are shown as black dots. ⇠ redshift Fig. 7. Values of coh determined for seven subsamples of the 6.1. ⇤CDM fit of the Hubble diagram Hubble residuals: low-zz < 0.03 and z > 0.03 (blue), SDSS z < 0.2 and z > 0.2 (green), SNLS z < 0.5 and z > 0.5 (orange), Using the distance estimator given in Eq. (4), we fit a ⇤CDM and HST (red). cosmology to supernovae measurements by minimizing the fol- lowing function:
2 1 = (µˆ µ (z; ⌦ ))†C (µˆ µ (z; ⌦ )) (15) may a↵ect our results including survey-dependent errors in es- ⇤CDM m ⇤CDM m timating the measurement uncertainty, survey dependent errors with C the covariance matrix of µˆ described in Sect. 5.5 and in calibration, and a redshift dependent tension in the SALT2 µ⇤CDM(z; ⌦m) = 5 log10(dL(z; ⌦m)/10pc) computed for a fixed model which might arise because di↵erent redshifts sample dif- 1 1 13 fiducial value of H0 = 70 km s Mpc , assuming an unper- ferent wavelength ranges of the model. In addition, the fit value turbed Friedmann-Lemaître-Robertson-Walker geometry, which of coh in the first redshift bin depends on the assumed value 1 is an acceptable approximation (Ben-Dayan et al. 2013). The of the peculiar velocity dispersion (here 150km s ) which is · free parameters in the fit are ⌦m and the four nuisance param- somewhat uncertain. 1 eters ↵, , MB and M from Eq. (4). The Hubble diagram for We follow the approach of C11 which is to use one value of the JLA sample and the ⇤CDM fit are shown in Fig. 8. We find coh per survey. We consider the weighted mean per survey of a best fit value for ⌦m of 0.295 0.034. The fit parameters are the values shown in Figure 7. Those values are listed in Table 9 given in the first row of Table 10.± and are consistent with previous analysis based on the SALT2 For consistency checks, we fit our full sample excluding sys- method (Conley et al. 2011; Campbell et al. 2013). tematic uncertainties and we fit subsamples labeled according to the data included: SDSS+SNLS, lowz+SDSS and lowz+SNLS. Confidence contours for ⌦m and the nuisance parameters ↵, 6. ⇤CDM constraints from SNe Ia alone and M are given in Fig. 9 for the JLA and the lowz+SNLS sample fits. The correlation between ⌦m and any of the nuisance The SN Ia sample presented in this paper covers the redshift parameters is less than 10% for the JLA sample. range 0.01 < z < 1.2. This lever-arm is su cient to provide The ⇤CDM model is already well constrained by the SNLS a stringent constraint on a single parameter driving the evolu- and low-z data thanks to their large redshift lever-arm. However, tion of the expansion rate. In particular, in a flat universe with the addition of the numerous and well-calibrated SDSS-II data a cosmological constant (hereafter ⇤CDM), SNe Ia alone pro- to the C11 sample is interesting in several respects. Most impor- vide an accurate measurement of the reduced matter density tantly, cross-calibrated accurately with the SNLS, the SDSS-II ⌦m. However, SNe alone can only measure ratios of distances, data provide an alternative low-z anchor to the Hubble diagram, which are independent of the value of the Hubble constant today 1 1 with better understood systematic uncertainties. This redundant (H0 = 100h km s Mpc ). In this section we discuss ⇤CDM parameter constraints from SNe Ia alone. We also detail the rel- 13 This value is assumed purely for convenience and using another ative influence of each incremental change relative to the C11 value would not a↵ect the cosmological fit (beyond changing accord- 1 analysis. ingly the recovered value of MB).
15 Brief reminder why the cosmological constant is regarded as a problem?
R = p g ⇢ The CC gravitates in General L 16⇡G vac Relativity: ✓ ◆ G = 8⇡G⇢ g µ⌫ vac µ⌫
Now: ⇢obs ⇢theory vac ⌧ vac
Just as well because anything much bigger than we have and the universe would have looked a lot different to what it does look like. In fact structures would not have formed in it.
3 Estimate what the vacuum energy should be :
⇢theory ⇢bare vac ⇠ vac + zero point energies of each particle
+ contributions from phase transitions in the early universe
4 zero point energies of each particle
For many fields (i.e. leptons, quarks, gauge fields etc...):
i 3 4 1 2 2 d k gi i < ⇥> = gi k + m 3 2 2 0 (2 ) 16 fields ⇥ ⇤ fields where gi are the dof of the field (+ for bosons, - for fermions).
5 contributions from phase transitions in the early universe
V (200 GeV)4 ewk ⇠
V (0.3GeV)4 QCD ⇠
6 Quantum Gravity cut-off (1018 GeV)4 fine tuning to 120 decimal places
SUSY cut-off (TeV)4 fine tuning to 60 decimal places EWK phase transition (200GeV)4 fine tuning to 56 decimal places QCD phase transition (0.3GeV)4 fine tuning to 44 decimal places Muon (100MeV)4
electron (1 MeV)4 fine tuning to 36 decimal places
(meV)4 Observed value of the effective cosmological constant today !
7 a˙ 2 8π k Λ Friedmann with Λ: H 2 ≡ = Gρ − + a2 3 a2 3 a(t) depends on matter.
€
w=1/3 – Rad dom: w=0 – Mat dom: w=-1– Vac dom
P w(a) = w +(1 a)w Typical parameterisation ⌘ ⇢ 0 a
Friedmann with evolving dark energy:
z 2 2 4 3 2 1 + w(z ) H (z)=H (1 + z) + (1 + z) + (1 + z) + exp 3 dz 0 r m k de 1 + z ⇤0 ⇥⇥
8 Planck Collaboration: Planck Cosmological Parameters
Planck Collaboration: Planck Cosmological Parameters
Fig. 32. The 2D joint posterior distribution betweenNe↵ and Yp with both parameters varying freely, determined from the Planck+WP+highL likelihood. The colour of each sample in Fig. 33. The 2D joint posterior distribution between Ne↵ and ✓s Markov chain indicates the associated value of ✓d/✓s. The Ne↵- from the LCDM+Ne↵+Yp (red) and LCDM+Ne↵ (blue) models, + + Fig.Yp relation 32. The from 2D the joint BBN posterior theory is distribution shown by the between dashedNe curve.↵ and using Planck WL HighL data. YThep with vertical both line parameters shows the varying standard freely, value determinedNe↵ = 3.046. from The the Planckregion+ withWPY+phighL> 0.294 likelihood. is highlighted The colour in gray of delineating each sample the re- in Fig. 33. The 2D joint posterior distribution between Ne↵ and ✓s Markovgion of the chain plot indicates exceeding the the associated 2 upper value limit of of✓ thed/✓ recents. The mea-Ne↵- from6.5. Dark the LCDM Energy+N Constraintse↵+Yp (red) and LCDM+Ne↵ (blue) models, Ysurementp relation of from initial the Solar BBN helium theory is abundance shown by (Serenelli the dashed & curve. Basu, using Planck+WL+HighL data. The2010 vertical). line shows the standard value N = 3.046. The A major challenge for cosmology is to elucidate the nature of the e↵ dark energy driving the accelerated expansion of the Universe. region with Yp > 0.294 is highlighted in gray delineating the re- gion of the plot exceeding the 2 upper limit of the recent mea- 6.5.The Darkmost Energy prosaic Constraintsexplanation is that dark energy is a cosmo- logical constant. An alternative is dynamicalPlanck dark Collaboration: energy mod- Planck Cosmological Parameters surementis thus an of approximate initial Solar degeneracy helium abundance between these(Serenelli two ¶me- Basu, 2010). Aels major (Wetterich challenge, 1988 for; Ratra cosmology & Peebles is to, elucidate1988), usually the nature based of the on ters. It can be partially broken by the phase shift of the acoustic darka scalar energy field. driving In the simplest the accelerated models expansion the field is of very the light, Universe. has a oscillations that arises due to the free streaming of the neutri- Planck+WP+BAO Planck+WP+SNLS 92 Thecanonical most kinetic prosaicPlanck+WP+Union2.1 energy explanation term and is that is minimally darkPlanck energy+WP coupled is a to cosmo- grav- 1.6 nos (Bashinsky & Seljak, 2004). The other, less important de- logicality. In such constant. models An the alternative dark energy is dynamical speed of dark sound energy equals mod- the 88 is thus an approximate↵ degeneracy between these↵ two parame- generacy breaking e ect, is the early ISW e ect discussed by elsspeed (Wetterich of light, and1988 it; Ratra has zero & Peebles anisotropicPlanck, 1988 Collaboration: stress.), usually It Planck thus based Cosmological con- on Parameters ters.Hou It et can al. ( be2011 partially). broken by the phase shift of the acoustic 84 atributes scalar field. very Inlittle the to simplest clustering. models We the shall field only is very consider light, has such a 0.8 oscillations that arises due to the free streaming of the neutri- 80 The joint posterior distribution between Ne↵ and Yp from the canonical1.0Planck kinetic+WP+BAO energy term and isPlanck minimally+WP+SNLS coupled to grav- 92 models in the following. H nosPlanck (Bashinsky+ + & Seljak, 2004). The other, less important de- a WP highL likelihood is shown in Figure 32 with the ity. In suchPlanck models+WP+Union2.1 the dark energyPlanck speed+WP of sound equals the 1.6 0
A simple way to parametrize dark energy is through its equa- w 88 76 generacy breaking e↵ect, is the early ISW e↵ect✓ discussed/✓ = / by 0.0 Planck Collaboration: Planck Cosmological Parameters colour of each MCMC sample coding the value of d s rd rs. speedtion of of state lightw andp it/⇢ has(Turner zero anisotropic & White, 1997 stress.). A It cosmolog- thus con- Hou et al. (2011). 0.8 84 The major constraint on Ne↵ and Yp comes from the precise mea- tributesical constant very haslittle⌘w to clustering.1 while We scalar shall field only models consider typically such 72 surementThe joint of this posterior ratio, leaving distribution the degeneracy between Ne along↵ and theYp from constant the ⌘ 0.8 modelshave1.0 time in thevarying following.w with w 1. The analysis2.1.4. performed Dark energy here 80 68 lensing likelihood, which encapsulates the lensing information Planck+WP+highL likelihood is shown in Figure 32 with the 0.8 H ✓d/✓s direction. The relation between Ne↵ and Yp from BBN a
A simple way to parametrize dark energy is through its equa- 0 is basedmax 0.6 on the “parameterized post-Friedmann” (PPF) frame- ✓ /✓ = / w 76 in the (mostly squeezed-shape) CMB trispectrum via a lensing colour of each MCMC sample coding31 the value of d s rd rs. P 0.0 64 theory is shown by the dashed curve . The standard BBN pre- tionwork of of/ stateHu &w Sawickip/⇢ ((2007Turner) and & WhiteHu In(2008, 1997 our) as). baseline A implemented cosmolog- model we assume that the dark energy is a cos- The major constraint on Ne↵ and Yp comes from the precise mea- 0.8P ⌘ potential power spectrum (Planck Collaboration 12, 2013). The diction with Ne↵ = 3.046 is contained within the 68% confi- ical constant has w 1 while scalar field models typically 1.6 72 surement of this ratio, leaving the degeneracy along the constant in CAMB (Fang et al.⌘ , 2008b,a) and discussedmological earlier in constant Sect. 2. with current density parameter ⌦⇤. When dence region. The gray region is for Y > 0.294 whichPlanck is Collaboration: the 2 Cosmologicalhave time0.4 parameters varying w with w 1. The analysis performed here 2.0 1.6 1.2 0.8 0.4 theoretical predictions for the lensing potential power spectrum p This allows us to investigate both regions of parameter space in 0.8 68 ✓d/✓s direction. The relation between Ne↵ and Yp from BBN considering a dynamical dark energy component,w we parame- max 0.6 0 conservative upper bound on the primordial31 helium abundance iswhich basedw is on less the than “parameterized minus one and post-Friedmann” models for which (PPF)w changes frame- are calculated by camb, optionally with corrections for the non- P Dark Energy 64 theory is shown by the dashed curve . The standard BBN pre- The CMB temperature data alone does not strongly constrain from (Serenelli & Basu, 2010). Most of the samples are consis- work/ of Hu & Sawicki (2007) and Hu terize(2008) as the implemented equation of state either as a constant w or as a function inw time.,P because of a strong geometrical degeneracy even for spatially- diction with Ne↵ = 3.046 is contained within the 68% confi- 0.2 1.6 linear matter power spectrum, along with the (non-linear) lensed tent with this bound. The inferred estimates of Ne↵ and Yp from in CAMB (Fang et al., 2008b,a) and discussed earlier in Sect. 2. Fig. 35. Plot illustrating the joint posterior for w and wa Y > . ► ParameterizeflatTo models.0.4 begin From wePlanck plot dark inwe Fig. find energy34 the marginalized usingof the PPF cosmological posterior framework prob- scale2.0 of factor, Hu1.6 anda1.2, withSawicki0.8 (2007)0.4 0 dencethe Planck region.+WP The+HighL gray region data are is for p 0 294 which is the 2 This allows us to investigate both regionsDark of parameter Energy space in for Planck , WMAP-polarization- and BAO data, marginalizingCMB power spectra. For the Planck temperature power spec- conservative upper bound on the primordial helium abundance abilities for models+0 with.62 w =constant. For these runs we have w0 which w isw less= 1 than.54 0 minus.50 (95% one, Planck and modelsTT+lowP) for, which(51)w changes 0.0 over other parameters. The contours are set at 68% and 95%.trum, corrections to the lensing e↵ect due to non-linear struc- from (Serenelli+0.59 & Basu, 2010). Most of the samples are consis-► takenNo anisotropic a flat prior on w stressesfrom 3 to 0.3. (Note thatp adding in N ↵ = 3.33 , (68% CL), (89a) in time.0.2 2.0 1.6 1.2 0.8 0.4 Independent flat priors of 3 < w < 0.3 and 2 < wa < 2 e 0.83 high-i.e.,` almostdata,Parameterise a not 2 shift illustrated, into the phantom results eos: domain. in littlew This( changea) is partly, to= thew poste-Fig.0 + 35.(1Plota illustrating)wa, the joint posterior for0 w0 and wa (4) tent with this bound. The inferred estimates of Ne↵ and Yp from To begin we plot in Fig. 34 thew marginalized posterior⇢ prob- ture growth can be neglected, however the impact on the lens- = . +0.041. . riors.)but not As entirely, expected, a parameterPlanck volumealone e↵ doesect, with not the strongly average⌘ ef- constrainfor wPlanck, were , WMAP-polarization- assumed. The colour and BAO of the data, scattered marginalizing points indicates the theYpPlanck0 254+WP0.+033HighL(68% data CL) are (89b) fective 2 improving by 2 2 compared to base ⇤CDM. halofit abilities0.0 for models with w =constant. For these runs weover have otherdistribution parameters. of The the contours Hubble parameter are set at 68%H0. Dashed and 95%. grey lines guideing potential reconstruction is important. We use the Fig.dueThis 34.to is the consistentPlot degeneracy indicating with theh of marginalized preference thisi⇡ parameter for a posterior higher with lensing the probabilities Hubble am- expan- for + . taken a flat prior2.0 on w1.6from 1.23 to 0.0.83. (Note0.4 that addingIndependent in the flateye priors to the of “cosmological3 < w0 < constant”0.3 and 2 solution.< wa < 2 With Y allowed0 59 to vary, N ↵ is no longer tightly constrained► sion.Degeneracyplitude We discussed can then in Sect.with attempt 5.1.2 H, to improving0 breakmeans the the fitand degeneracyPlanck in the assume w < alone by1 combin- that can the darkonly energy weakly does constrain not interact dark with energy other con- model (Smith et al., 2003) as updated by Takahashi et al. (2012) Ne↵ = 3P.33 0.83, (68% CL)e , (89a) Planckthe dark` energyalone equation-of-state weak constraintsw parameter on wDE(assumed because con- of degeneracy of w with H0: high- data, not illustrated, results in little change to the poste-were assumed. The colour of the scattered points indicates the by the value+ of0.041✓d/✓s. Instead, it is constrained due, at least in ingregion,Planck wherewith the lensing other datasets. smoothing Adding amplitude in becomes BAO dataslightly tightens the to model the impact of non-linear growth on the theoretical pre- stant),riors.) As for expected, the indicatedPlanck combinationsalone does of notstituents data strongly sets. constrain Aother flatdistribution prior thanw, through of the Hubble gravity. parameter SinceH0. Dashed this grey model lines guide allows the Yp = 0.254 0.033. (68% CL). (89b)► larger. However, the lower limit in Eq. (51) is largely determined Planck Collaboration: Cosmological parameters part, to the impact that varying Ne↵ has on the phase shifts of the Fig.posteriorCan 34.w Plot be probability, indicating broken marginalized. giving by CMB posterior 1lensing1 probabilities (see for later) and other probes Breakondue tofrom the with degeneracy3 to other0 3 wasof probes< this assumed. parameter including The with dashed the lensing, Hubblegrey line expan-the indi-SN, eye toBAO the “cosmological ... constant” solution. diction for the lensing potential power. the darkby the energy (arbitrary) equation-of-state prior H0 100 parameter km s Mpc wequation, chosen(assumed for the con- of state to cross below 1, a single-fluid model can- acoustic oscillations. As shown in Hou et al. (2012b), this e↵ect catesHubble the parameter. “cosmological Much of constant” the posterior solution. volume in the phan- 1.6 With YP allowed to vary, Ne↵ is no longer tightly constrained sion. We can then attempt to break the degeneracy by combin-2 grees of freedom, models with a combination of the two, and explains the observed correlation between Ne↵ and ✓s. This cor- stant),wtom= for region1. the13 indicated is0 associated.24 combinations(95% with, Planck extreme of values+ dataWP fornot sets.+ BAO) cosmological beA flat, used prior self-consistently.(90) We therefore use the parameterized by the value of ✓d/✓s. Instead, it is constrained due, at least in►onExampleingw fromPlanck 3 to±with - if0. other3 assume was datasets. assumed. wa Adding The = dashed0, 95% in BAO grey CL line data indi- tightens the Planck TT+lowP+ext models with massive sterile neutrinos. In each subsection we relation is shown in Fig. 33. The correlation in the ⇤CDM+Ne↵ Settingparameters,which w a = are excluded0 obtain by other astrophysicalpost-Friedmann data. The (PPF) model of Fang et al. (2008a).emphasize For models the Planck-only constraint, and the implications of part, to the impact that varying Ne↵ has on the phase shifts of the catesposteriormild the tension“cosmological probability, with base constant”⇤ givingCDM disappears solution.⇤ as we add more data 1.6 Planck TT+lowP+WL 2.2. Parameter choices model is also plotted in the figure showing that the Ne↵-Yp de- stillin good basically agreement consistent with with the aCDM cosmological model. Using constant, supernovae though 0.8 the Planck result for late-time cosmological parameters mea- acoustic oscillations. As shown in Hou et al. (2012b), this e↵ect with w > 1,1 the PPF model agreesPlanck withTT+lowP+WL+ fluidH models0 to signif- generacy makes the phase shift e↵ect much more significant. SNLSdatathat leads break does to the lead the geometrical stronger to a slightly degeneracy. constraints lower Adding valuePlanck of wlensingthan Union2.1. sured from other observations. We then give a brief discussion of explains the observed correlation between N ↵ and ✓ . This cor- wand= BAO,1.13 JLA0 and.24H0 (95%(“ext”), givesPlanck the+ 95WP % constraints:+ BAO), (90) e s + icantly better accuracy thana required for the resultstensions reported between inPlanck and some discordant external data, and Fig. 27. Samples from the distribution of the dark energy pa-stillIf basically instead we consistent± combine withPlanck a cosmologicalWP with constant, HST measurements though of 0.8 2.2.1. Base parameters relation is shown in Fig. 33. The correlation in the ⇤CDM+N ↵ +0.091 w 31 rameters w0 and wa using Planck TT+lowP+BAO+JLAe data, Hww,= the= 11 di..09023↵erence0.17Planck between (95%TT,+ thePlancklowP values+ext+ ;WP of H+ Union2preferred(52a).1), by CMB(91) 0.0 assess whether any of these model extensions can help to resolve For constant Ne↵ , the variation due to the uncertanty of the baryon SNLS0 does lead0 to.096 a slightly lower⇤ value of wthisthan0 paper. Union2.1. 0 model is also plottedcolour-coded in the by the figure value showing of the Hubble that parameter the Ne↵H-Y. Contoursp de- in good agreement± with the CDM model. Using supernovae them. Finally we provide constraints on neutrino interactions. 0 ++00..13085 a Ifand instead HST we reflects combine itselfPlanck in+ theWP joint with constraint HST measurements of ofa density is too small to show given the thickness of the curve. ww== 11..13006 . Planck(95%,TTPlanck+lowP+lensingWP ++extSNLS) ; , (52b) (92) The first section of Table 1 lists our base parameters that have show the corresponding 68 % and 95 % limits. Dashed grey lines data leads to0 the0.14091 stronger constraints w ↵ w generacy makes the phase shift e ect much more significant. H , the di ↵erence between the values of H preferred by CMB 0.0 0 +0+.180.075 0 Planck+WP+BAO intersect at the point in parameter space corresponding to a cos- w = 1.019 0.080 Planck TT, TE, EE+lowP+lensing+ext . flat priors when they are varied, along with their default values andw HST= 1 reflects .24 0 .19 itself(95% in the, Planck joint constraint+ WP + ofHST), 1 (93) 0.8 6.4.1. Constraints on the total mass of active neutrinos 31 mological constant. w = 1.09 0.17 (95%, Planck + WP + Union2.1), (91) Planck+WP+Union2.1 For constant Ne↵ , the variation due to the uncertanty of the baryon (52c) +0.18 ± 2.1.5. Power spectra47 Planck+WP+BAO in the baseline model. When parameters are varied, unless oth- w which= 1.24 is in tension+0(95%.13 , withPlanckw+=WP1.+ HST), (93) Planck+WP+SNLS Detection of neutrino oscillations has proved that neutrinos have density is too small to show given the thickness of the curve. w = 10..1319 0.14 (95%, Planck + WP + SNLS), (92) 0.8 The addition of Planck lensing, or using the full Planck tem- Planck+WP+Union2.1 mass (see e.g., Lesgourgueserwise & Pastor stated,2006, for prior a review). ranges The are chosen to be much larger than the ► peratureIf w ,+polarization1 then it likelihood is likely totogether change with with the time.BAO, JLA, In order2 to in- 1.6 whichMild is in tension tension with forw = w<-11. but notOver the last decades therePlanck +WP+SNLS has been significantPlanck progressbase ⇤CDM in model assumes a normal mass hierarchy vestigateand H data this does we consider not substantially a linear improve model, thew( constrainta) = w0 + ofwa(1 a), 2.0 1.6 1.2 0.8 0.4 posterior, and hence do not a↵ect the results of parameter esti- This constraint is unchanged at the quoted precision if we add If w , 0 1 then it is likely to change with time. In order to in- 1.6 with m⌫ 0 .06 eV (dominated by the heaviest neutrino mass significantEq. (52aw ). All of these data set combinations areimproving compatible with w the accuracy,47 speed and generality of the numerical⇡ the JLA supernovae data and the H0 prior of Eq. (30). vestigatewhere this0 is we the consider value ofa linear the equation model, w of(a) state= w today+ w (1 anda), a deter- 2.0 1.6 1.2 0.8w0 eigenstate)0.4 but there are other possibilities including a degen- ⇤ Planck 2015= : 0 a 3 mation. In addition to these priors, we impose a “hard” prior on Figure 26 illustrates these results in the ⌦ –⌦ plane. We minesthe base howCDM the equation value of w of state1. In PCP13 evolves, we away conservatively from w near the erateP hierarchy with m⌫ > 0.1 eV. At this time there are no m ⇤ where w0 is the value+ of0.24 the equation of state todaycalculation and wa deter-0 of the2 CMB power1 spectraw00 given1 an ionization 1 1 quoted w = 1.13 , based on combining Planck with BAO, the⇠ Hubble constant of [20, 100] km s Mpc . adopt Eq. (50) as our most reliable constraint on spatial curva-►minespresentWith how epoch thevariable equation (Chevallier0.25 ofw(a) state & evolves Polarskisimilar away, 2001 conclusion from; Linderw near, 2003 the ). This compelling theoretical reasons to prefer strongly any of these as our most reliable limit on w. The errors in Eqs.history (52a)–(052c) are and set ofFig. 36. cosmologicalPlot illustratingw0 parameters the joint posterior (Sugiyama for w0 and, Pwa, ture. Our Universe appears to be spatially flat to an accuracy ofpresentparametrization epoch (Chevallier captures & Polarski the low-redshift, 2001; Linder behaviour, 2003). This of our mod- possibilities,↵ so allowing for larger neutrino masses is perhaps 0.5%. substantially smaller, mainly because of the addition1995 of; theMa JLA & BertschingerFig. 36. Plotmarginalizing illustrating, 1995 over the; other jointSeljak parameters,posterior & Zaldarriaga for forw0 dioneanderent ofw thea,, choices1996 most well-motivated; of ad- extensions to base ⇤CDM con- parametrizationelsSNe (light data, minimally-coupled which captures o↵er a the sensitive low-redshift scalar probe of fields) behaviour the dark as energy well of our as equa- mod- many others 9 How should we Fig.parameterise 28.marginalizingMarginalizedditional posteriorw? over data other distributions to parameters,Planck forand for (w WMAP-polarization.0 di, w↵aerent) for var- choicessidered of ad- in Contour this paper. levels There has also been significant interest els (light minimally-coupledz < scalar fields) as well as many others 18 astion long of as state they at do1. not In contributePCP13, the addition significantly ofSeljak the SNLS to the, SNe totalious1996 data energyditional; combinations.White dataare to set &Planck We at Scott 68% showandPlanck and, WMAP-polarization.1996 95%.TT+lowP The; Hu grey in combi- & contours Contour Whiterecently levels use, BAO.in1997 larger The; neutrino red2.2.2. masses Derived as a possible parameters way to lower 6.3. Dark energy as longdata as pulled theyw dointo not⇠ the contribute phantom domain significantly at the 2 to level, the total reflecting energy density at early times. The dynamical evolutionZaldarriagaNottingham, of w(nationa) can et withareMarch al.lead set BAO,, at19982013contours 68% JLA,; andH0Hu(“ext”), use 95%. et Union2.1 The al. and, grey two1998 supernovae data contours combinations; Bucher use data. BAO. The et The8, al. the blue red, late-time2000 contours fluctuation; use amplitude, and thereby reconcile densitythe tensionat early between times. The the SNLS dynamical sample evolution and the Planck of w(2013a) can base leadwhich add the CFHTLenS data with ultra-conservative cuts as The physical explanation for the observed accelerated expansion to⇤ distinctive imprints in the CMB (Caldwell et al., 1998) whichcontoursSNLS use Union2.1 supernovae supernovae data. data. Dashed The grey blue contourslinesPlanck guide usewith the weak eye lensing to theMatter-radiation measurements and the abundance equality of zeq is defined as the redshift at which to distinctiveCDM parameters. imprints in As the noted CMB in Sect. (Caldwell5.3, this et discrepancyHu al., ,19982000) is which nodescribed; Lewis in the & text Challinor (denoted “WL”)., 2002 Dashed; Seljak grey lines et show al., 2003; Doran, of the Universe is currently not known. In standard ⇤CDMWednesday, the wouldlonger 20 show present,March up following13 in the Planck improveddata. photometric calibrations of SNLS supernovae data. Dashed grey lines guide the eyerich to clusters the (see Sects. 5.5 and 5.6). Though model dependent, would show up in the Planck data. the parameter values“cosmological corresponding constant” to a cosmological solution. constant. ⇢ + ⇢⌫ = ⇢c + ⇢b (where ⇢⌫ approximates massive neutrinos as acceleration is provided by a cosmological constant satisfying an theIn SNe Fig. data35 inwe the plot JLA contours sample. One of consequence the joint2005 posterior of this; Challinor is probabilitiesthe “cosmological & Lewis constant”, 2005 solution.; Cyr-Racine & Sigurdsonneutrino, mass2011 constraints; from cosmology are already signifi- equation of state w pDE/⇢DE = 1. However, there are many In Fig. 35 we plot contours of the joint posterior probabilities⇤ cantly stronger than those from tritium beta decay experiments ⌘ fortighteningw0 and ofwa theusing errorsPlanck in Eqs. (+52aWP)–(+52cBAO) around data. the WeCDM use indepen- massless). possible alternatives, typically described either in terms of extrafor wvalue0 andww=a using1 whenPlanck we combine+WP the+BAO JLA data. sampleBlas We with usePlanck et indepen- al.. , 2011; Lesgourgues & Tram, 2011; Howlett(see e.g., Drexlin et al. et, al. 2013). dent flat priors of 3 < w0 < 0.3 and 2 < wa < 2. The 10 degrees of freedom associated with scalar fields or modificationsdent flatIf priorsw di↵ers of from3 < 1,w0 it< is likely0.3 toand change2 < withwa time.< 2. We The Herecamb we give constraints assumingThe redshift three species of of recombination, degener- z , is defined so that the op- 2012 ). Our⇤ baselinethe choicethe of choice numericaldataset of used dataset for theBoltzmann used SNe, for and the this SNe, continuescode and is this with continues with ⇤ of general relativity on cosmological scales (for reviews see e.g.,pointspointsconsider are are coloured here coloured the case by theof by a Taylor value the value expansion of H0 of, which ofH0w,at which shows first order a shows clear inCDM a clear predictions. This tension can be seen even in the sim- ate massive neutrinos, neglecting the small di↵erences in mass Copeland et al. 2006; Tsujikawa 2010). A detailed study of these ple modelthe of dark Eq. energythe (53). dark The parameters. energy green regions parameters. The results in Fig. for28 TheshowPlanck results 68+ %Union2.1 for Planck are+Union2.1 aretical depth to Thomson scattering from z = 0 (conformal time variationvariationthe scale with factor, withw0 and parameterizedw0 andwa. Thewa. by “cosmological The “cosmological(March constant” constant” point 2013; pointLewis et al., 2000), a parallelizedexpected line-of-sight from the observed mass splittings. At the level of sensi- models and the constraints imposed by Planck and other data is and 95 %in contours better agreementin in better the w0– agreementw witha plane a cosmological for withPlanck aTT cosmological constant+lowP com- than constant thosetivity of forPlanck than thosethis is an for⌘ accurate= ⌘ approximation,) to z = z butis note unity, that it assuming no reionization. The optical (w(0w, w0,a)w=a)(=1(, 0)1 lies, 0) withinlies within the 1 the contour 1 contour andcode the and marginal- developed the marginal- from cmbfast (Seljak & Zaldarriaga, 1996) 0 presented in a separate paper, Planck Collaboration XIV (2015). w = w0 + (1 a)wa. (53)bined withPlanck the+ CFHTLenSSNLS.Planck We+SNLS. H13 remark data. We that In remark this the example, variations that thewe in have variations the constraintsdoes not in the quite constraints match continuously on to the base⇤⇤CDM model izedized posteriors posteriors for w for0 andw wanda arew are Here we will limit ourselves to the most basic extensions 0 a and Cosmicsappliedon “ultra-conservative”(Bertschinger dark energyon dark parameters energy cuts,, 1995 excluding parameters using; Ma di⇠↵erententirely & using combinations Bertschinger and di↵ ex-erent(which combinations of data, assumes1995), two of data masslessdepth and is one given massive by neutrino with + . of ⇤CDM, which can be phenomenologically described in More complex0 72+0 models.72 of dynamical dark energy are discussedcludingsets measurements might be due with to✓ unmodelled< 170 in ⇠+ systematicsfor all tomographic in the analysis,m the= 0.06 eV). We assume that the neutrino mass is con- w0 = =1.04. 0.69 (95%, Planck, + WP++ BAO)+, , (94) sets might be due to unmodelled systematics in⌫ the analysis, the terms of the equation of state parameter w alone. Specifically w0in Planck1 04 Collaboration0.69 (95% XIV Planck(2015). FigureWP27whichBAO)shows the calculates 2D (94) the lensed CMB temperature and polariza- ⌘ redshiftpotential bins. As discussed presence in ofPlanck which Collaboration have been discussed XIV (2015 in), Sects.stant,5.3 and that the distribution function is Fermi-Dirac with zero we will use the camb implementation of the “parameterizedwa
7 The acceleration has not been forever -- pinning down the turnover will provide a very useful piece of information.
Huterer 2010
Help address cosmic coincidence problem ! A region hopefully DES and EUCLID will be able to probe 10 Approaches to Dark Energy:
! A true cosmological constant -- but why this value? ! Time dependent solutions arising out of evolving scalar fields -- Quintessence/K-essence. ! Modifications of Einstein gravity leading to acceleration today. ! Anthropic arguments. ! Perhaps GR but Universe is inhomogeneous. ! Hiding the cosmological constant -- its there all the time but just doesn’t gravitate ! Yet to be proposed ...
11 String - theory -- where are the realistic models? `No go’ theorem: forbids cosmic acceleration in cosmological solutions arising from compactification of pure SUGR models where internal space is time- independent, non-singular compact manifold without boundary --[Gibbons] Avoid no-go theorem by relaxing conditions of the theorem. 1. Allow internal space to be time-dependent scalar fields (radion) 2. Brane world set up require uplifting terms to achieve de Sitter vacua hence accn Example of stabilised scenario: Metastable de Sitter string vacua in TypeIIB string theory, based on stable highly warped IIB compactifications with NS and RR three- form fluxes. [Kachru, Kallosh, Linde and Trivedi 2003] Metastable minima arises from adding positive energy of anti-D3 brane in warped V Calabi-Yau space. 0.5 V s 100 150 200 250 300 350 400 1.2 -0.5 1 0.8 -1 0.6
-1.5 0.4 0.2 -2 s 100 150 200 250 300 350 400 AdS minimum Metastable dS minimum 12 The String Landscape approach Type IIB String theory compactified from 10 dimensions to 4. Internal dimensions stabilised by fluxes. Assumes natural AdS vacuum uplifted to de Sitter vacuum through additional fluxes !
500 -500 Many many vacua ~ 10 ! Typical separation ~ 10 Λpl Assume randomly distributed, tunnelling allowed between vacua --> separate universes .
-118 Anthropic : Galaxies require vacua < 10 Λ pl [Weinberg] Most likely to find values not equal to zero! Landscape gives a realisation of the multiverse picture. There isn’t one true vacuum but many so that makes it almost impossible to find our vacuum in such a Universe which is really a multiverse. So how can we hope to understand or predict why we have our particular particle content and couplings when there are so many choices in different parts of the universe, none of them special ? 13 SUSY large extra dimensions and Lambda - Burgess et al 2013, 2015 Soln to 6D Einstein-Maxwell-scalar with chiral gauged sugr. In more than 4D, the 4D vac energy can curve the extra dimensions. Proposal: Physics is 6D above 0.01eV scale with SUSY bulk. We live in 4D brane with 2 extra dim. 4D vac energy cancelled by Bulk contributions - quintessence like potential generated by Qu corrections leading to late time accn.
SequesteringOur Lambda proposal - Kaloper and Padilla 2013-2016 IR soln to the problem - initial version adds a global term to Einstein action
Introduce global dynamical variables Λ, λ
2 4 Mpl 4 2 µ S = d x g R ( g , ) + 2 L 4µ4 Padilla 2015 λ sets the hierarchy between matter scales and Mpl m m phys = Mpl Mpl
14 Equations of motion
Equations of motion 2 µ µ 1 µ ↵ MplG ⌫ = ⌧ ⌫ ⌫ ⌧ ↵ Eq of motion: 4 h i
T µ = V µ + ⌧ µ ⌫ vac ⌫ ⌫ 1 M 4 Gµ = µ + µ Universe has finite spacetimepl volume 4 ⇤ equation : 4 1 ↵ d xQpg 0 where: ⇤ = =T d↵4pg, Q = spacetime volume must be finite 4µ44h i h i d4xpg Z R equation : space-time volume must be finite or else 0 ! mphys m R Vacuum energy= drops out at each and every loop order M2pl Mµpl µ µ gµ⌫ equation : Mif plG0 particle⌫ = masses go⇤! to zero⌫ + T⌫ ! UniverseNo hidden has equations finite spacetime — this is volumeeverything! EndsCOLLAPSE in a crunch TRIGGER = 1DARK ENERGY w=-1 is transient Residual cosmologicalcollapse constant triggered effby dominating= dark energy Ωk>0 4
3 form protected by shift symmetry, Linear potential V=m φ size of m3 technically natural
New local version of sequestering can accommodate infinite universe [Kaloper15 et al 2015] with Charmousis, Padilla and Saffin Self tuning - with the Fab Four PRL 108 (2012) 051101; PRD 85 (2012) 104040 In GR the vacuum energy gravitates, and the theoretical estimate suggests that it gravitates too much. Basic idea is to use self tuning to prevent the vacuum energy gravitating at all. The cosmological constant is there all the time but is being dealt with by the evolving scalar field.
Most general scalar-tensor theory with second order field equations:
[G.W. Horndeski, Int. Jour. Theor. Phys. 10 (1974) 363-384] The action which leads to required self tuning solutions :
In other words it can be seen to reside in terms of the four arbitrary potential functions of ϕ coupled to the curvature terms. Covers most scalar field related modified gravity models studied to16 date. fab four cosmology
“radiation” “matter” aa¨ 1.4 0.7 q = 2 a˙ 1.2 0.6 p 1/h a t t 1 0.5 ⇠ ⇠ q(t) 0.8 0.4 p(p 1) q(t) q = = (1 + h) 0.6 p2 0.3 0.4 0.2
0.2 0.1
0 −6 −4 −2 0 0 −8 −6 −4 −2 0 10 10 10 10 10 10 10 10 10 a(t)/a(t ) a(t)/a(t ) f f
Thursday, See28 February also: 2013 Appleby et al JCAP 1210 (2012) 060; Amendola et al PRD 87 (2013) 2, 023501; Martin-Moruno et al PRD17 91 (2015) 8, 084029; Babichev et al arXiv:1507.05942 [gr-qc] , Particle physics inspired models of Dark Energy ? Pseudo-Goldstone Bosons -- approx sym φ --> φ + const. Leads to naturally small masses, naturally small couplings
Barbieri et al
4 V (⇥)= (1 + cos(⇥/Fa)) Axions could be useful for strong CP problem, dark matter and dark 18 energy - see also recent work by D’Amico, Hamil & Kaloper 2016. Axions could be useful for strong CP problem, dark matter and dark energy. 2 QCD Strong CP problem intro axion : ma = ; Fa decay constant Fa
PQ axion ruled out but invisible 9 12 axion still allowed: 10 GeV Fa 10 GeV Sun stability CDM constraint String theory has lots of antisymmetric tensor fields in 10d, hence many light axion candidates.
17 18 Can have Fa ~ 10 -10 GeV
Quintessential axion -- dark energy candidate [Kim & Nilles].
18 Requires Fa ~ 10 GeV which can give:
3 4 33 E = (10 eV) m 10 eV vac ⇥ axion Because axion is pseudoscalar -- mass is protected, hence avoids fifth force constraints 19 Slowly rolling scalar fields -- Quintessence Peebles and Ratra; Wetterich; Ferreira and Joyce Zlatev, Wang and Steinhardt
Dashed line - radiation and matter Solid line - Quintessence enters tracking regime (4) and dominates (5)
Nunes Attractors make initial conditions less important 20 Scaling for wide range of i.c.
Fine tuning:
Generic issue Fifth force - Mass: require screening mechanism! 21 Screening mechanisms
1. Chameleon fields [Khoury and Weltman (2003) …] Non-minimal coupling of scalar to matter in order to avoid fifth force type constraints on Quintessence models: the effective mass of the field depends on the local matter density, so it is massive in high density regions and light (m~H) in low density regions (cosmological scales).
2. K-essence [Armendariz-Picon et al …] Scalar fields with non-canonical kinetic terms. Includes models with derivative self-couplings which become important in vicinity of massive sources. The strong coupling boosts the kinetic terms so after canonical normalisation the coupling of fluctuations to matter is weakened -- screening via Vainshtein mechanism
Similar fine tuning to Quintessence -- vital in brane-world modifications of gravity, massive gravity, degravitation models, DBI model, Gallileons, ....
3. Symmetron fields [Hinterbichler and Khoury 2010 ...] vev of scalar field depends on local mass density: vev large in low density regions and small in high density regions. Also coupling of scalar to matter is prop to vev, so couples with grav strength in low density regions but decoupled 22 and screened in high density regions. New quantum version [Burrage,EC, Millington 2016] 4. Interacting Dark Energy [Kodama & Sasaki (1985), Wetterich (1995), Amendola (2000) + many others… ] Ex: Including neutrinos -- 2 distinct DM families -- resolve coincidence problem Amendola et al (2007) Depending on the coupling, find that the neutrino mass grows at late times and this triggers a transition to almost static dark energy. Trigger scale set by time when neutrinos become non-rel
23 mν Perturbations in Interacting Dark Energy Models [Baldi et al (2008), Tarrant et al (2010)] Perturb everything linearly : Matter fluid example ⇤˙ 3 ⇥¨ + 2H 2 ⇥˙ H2[(1 + 2 2) ⇥ + ⇥ ]=0 c M c 2 c c b b ⇥ modified vary DM extra grav particle friction interaction mass Include in simulations of structure formation : GADGET [Springel (2005)] EAGLE [Virgo consortium] Halo mass function modified.
Halos remain well fit by NFW profile.
Density decreases compared to ΛCDM as coupling β increases.
Scale dep bias develops from fifth force acting between CDM particles. enhanced as go from linear to smaller non- linear scales.
Still early days -- but this is where I think there should be a great deal of development (Puchwein et24 al 2013, Density decreases as coupling β increases Barreira et al 2014, Arnold et al 2016) The problem of coupling DE and DM directly with scalars [D’Amico, Hamil & Kaloper 2016] Generate loop corrections to the DE mass.
Consider Yukawa type coupling between ¯ DE scalar and DM fermion g 33 Now since it is DE: m H 10 eV ' ⇠ Very light so long range Pot : (r) g2/r attractive 5th force: ⇠ Must be less than grav attraction of g m2 g2m2 Two tests [Burrage, Davis, Shaw 2008,2009] Look for evidence of DE through changes in the scatter of luminosities of high energy sources. Look for evidence of correlation between poln and freq of starlight . 26 Dark Energy Direct Detection Experiment [Burrage, EC, Hinds 2015,Hamilton et al 2015] Atom Interferometry Idea: Individual atoms in a high vacuum chamber are too small to screen the chameleon field and so are very sensitive to it - can detect it with high sensitivity. Can use atom interferometry to measure the chameleon force - or more likely constrain the parameters ! ⇤2 ⇢ 2 2 GMAMB MP = 2 + F = 1+2 r M r r2 A B M " ✓ ◆ # 2 2 i = 1 for ⇢iRi < 3M bg Log acceleration g 3M 10 = bg for ⇢ R2 > 3M 1 i 2 i i bg ⇢iRi H ê L M eV 0 3 - Sph source A and test object B 10 ë 0 L I near middle of chamber 10 -5 Log experience force between them -1 -10 - usually ƛ<<1 in cosmology but for atom ƛ=1 - reduced ---640246 -2 27 -14 -12 -10 -8 -6 -4 -2 0 suppression Log10 M Mp Figure 2: Avaliable parameter space ⇤ M for the chameleon. Atomic deflection experiments18, 19 H ê L exclude the top left region above the dotted line. The contour plot shows the acceleration of a rubidium atom in the vacuum chamber of Fig. 1, due to the chameleon force outside a sphere A having A < 1. Here we take RA =1cm and normalise the acceleration to the g of free fall on earth. The heavy solid line shows how far a first atom interferometer experiment can penetrate into this parameter space, while the heavy dashed line shows how far one can expect the measurement to be extended with attention to systematic errors. For ⇤ 1 meV, as suggested by the Planck ' survey8, atom interferometry should be able to detect chameleon physics up to the mass scale 2 16 M 2 10 M =5 10 GeV. ' ⇥ P ⇥ being insignificant in comparison. Above the dashed line in Fig. 1, B =1for a caesium atom. The dotted line is for lithium atoms. Atoms in high vacuum have already been used to measure gravitational forces with high precision, e.g.16, 17, but with source masses that are outside the vacuum chamber. Because of the intervening vacuum wall, the chameleon field within the chamber is essentially unaffected by the external source, in close analogy with Faraday shielding in electrostatics, as we discuss more fully in the supplementary material. Consequently, these experiments place no useful constraints on the chameleon parameters. By contrast, measurements of the van der Waals18–20 and Casimir-Polder21, 22 forces on in- dividual atoms use macroscopic sources inside the vacuum. However, the more recent, and more sensitive, measurements cannot detect the chameleon field because they use plane sources, which 4 Modifying Gravity rather than looking for Dark Energy - non trivial Any theory deviating from GR must do so at late times yet remain consistent with Solar System tests. Potential examples include: •f(R), f(G) gravity -- coupled to higher curv terms, changes the dynamical eqns for the spacetime metric. Need chameleon mechanism [Starobinski 1980, Carroll et al 2003, Hu and Sawicki 2007,…] • Modified source gravity -- gravity depends on nonlinear function of the energy. • Gravity based on the existence of extra dimensions -- DGP gravity We live on a brane in an infinite extra dimension. Gravity is stronger in the bulk, and therefore wants to stick close to the brane -- looks locally four-dimensional. Tightly constrained -- both from theory [ghosts] and observations • Scalar-tensor theories including higher order scalar-tensor lagrangians -- recent examples being Galileon models • Massive gravity - single massive graviton bounds m>O(1meV) from demand perturbative down to O(1)mm - too large to conform with GR at large distances 28 [Burrage et al 2013] Testing models - consider coupled dark energy-dark matter. Have seen provides a nice way to explain coincidence problem. What is most general phenomenological model we can construct? Three distinctCoupled classes CDM/DE of mixed models models with couplings intro at the level of the action [Pourtsidou, Skordis , EC 2013, 2015] Consider Dark Energy (DE) coupled to Cold Dark Matter (c) [e.g. Kodama & Sasaki ’84, Ma & Bertschinger ’95] T (c) and T (DE) are not separately conserved: T (c)µ = T (DE)µ = J =0 rµ ⌫ rµ ⌫ ⌫ 6 Various forms of coupling have been considered. Examples: J ⇢ [Amendola ’00] ⌫ / cr⌫ J ⇢ u(c) [Valiviita et al ’08] ⌫ / c ⌫ FRW background with J¯⌫ =(J¯0, J¯i) and linear perturbations ( J0, Ji). Note that J¯i =0because of isotropy. The CDM energy density equation becomes ⇢¯˙ +3 ⇢¯ = J¯ 29 c H c 0 With thanks to my collaborator Alkistis Pourtsidou for lending me some of the following slides. Alkistis Pourtsidou Swansea University, March 17th 2015 Models of Dark Matter coupled to Dark Energy FluidsUsing in General the fluid Relativity pull back formalism we consider the fluid/particle number density n. The action for GR and a fluid is 1 S = d4xp gR d4xp gf(n) 16⇡G Z Z f(n) is (in principle) an arbitrary function, whose form determines the equation of state and speed of sound of the fluid For pressureless matter (CDM) f(n) n / Stress-energy tensor is given by Tµ⌫ =(⇢ + P )uµu⌫ + Pgµ⌫ Coupled Lagrangian: general case Can match ⇢,P to the fluid function f(n) as df [AP, Skordis & Copeland (2013)] ⇢ = f, P = n f ) dn We want to construct a model where the fluid with number density Alkistis Pourtsidou Swansea University, March 17th 2015 Models of Dark Matter coupled ton Dark(e.g. Energy CDM) is explicitly coupled to a DE field Invariants: Y = 1 ( )2, Z = uµ 2 rµ rµ Our general Lagrangian has the form L = L(n, Y, Z, ) Example: Usual quintessence has L = Y + V ( )+f(n) 30 We can now consider di↵erent classes of theories by “breaking” the general Lagrangian we constructed in di↵erent ways. Alkistis Pourtsidou Swansea University, March 17th 2015 Models of Dark Matter coupled to Dark Energy Type 1 Type-1 models are classified via L(n, Y, Z, )=F (Y, )+f(n,Type ) 1 Type 1 f(n)=g(n)e↵( ) Type-1 models are classified via L(n, Y, Z, )=F (Y, )+f(n, ) Type-1 modelsNote are there classified is no Z viadependence. Type 1 models. L(n, Y, Z, )=F (Y, )+f(n, ) ex: f(n)=g(n)e↵( ) These models describe a K-essence scalar field coupled to matter. If F = Y + V ( ), we describe coupled quintessence models. Could fbe(n)= K-essenceg(n)e↵( ) scalar field coupled to matter, or QuintessenceNote there is no if ZF=Y+V(dependence.ϕ) d↵( ) Coupling current Jµ = ⇢ µ [generalized Amendola model] Note there is no Z dependence. d r These models describe a K-essence scalar field coupled to matter. If F = Y + V ( ), we describe coupled quintessence models. Choose ↵( )=↵ with ↵ const and study observational These models describe a K-essence0 scalar0 field coupled to matter. If signatures in CMB and matter power spectra (modified CAMB code). F = Y + V ( ), we describe coupled quintessence models. d↵( ) Type 1: Evolution of ⌦cdm Coupling current Jµ = ⇢ d µ [generalized Amendola model] 3 ↵( ) r Note the evolutiond↵ of( ) CDMType density: 1:⇢¯ Matterc =¯⇢c,0a powere spectra Coupling current Jµ = ⇢ d µ [generalized Amendola model] Alkistis Pourtsidou Swansea University, Marchr 17th 2015 Models of Dark Matter coupled to Dark EnergyChoose ↵( )=↵0 with ↵0 const and study observational 1 signatures in CMB and matter power spectra (modified CAMB code). α =0 0 Choose ↵( )=↵ with ↵ αconst0 = 0 and study observational10000 0 0 α =0.15 α = 0.15 0 ΩC 0 P(k) 0.8 signatures in CMB and matter power spectra (modified CAMB code). 3 ↵( ) Note the evolution of CDM density: ⇢¯c =¯⇢c,0a e ) 3 100 0.6 3 ↵( ) Alkistis Pourtsidou Swansea University, March 17th 2015 Models of Dark Matter coupled to Dark Energy Mpc Note the evolution of CDM density: ⇢¯ =¯⇢ a e -3 c c c,0 Ω (k) (h 0 0.4Alkistis Pourtsidou Swansea University, March 17th 2015 Models of Dark Matter coupled toP Dark1 Energy 0.2 0.01 0 0.0001 0.01 1 100 0.4 0.6 0.8 1 -1 a k (h Mpc ) P (k) a↵ected on small scales. There is more dark matter at early times, TheAlkistisMore CDM Pourtsidou density SwanseaDM is University, higher at March early at 17th early 2015 timestimes,Models for of Dark the Matterequality coupled coupled to case, Dark Energy earlier in ordermatter-radiation - only equalitysmall earlier.scale Only pertns small scalehave perturbations time to have to evolve to the same cosmological parameters today. The time to enter the horizon and grow during radiation-dominated era. The matter-radiationenter horizon equality occurs and earlier grow in the coupledduring case. radiation dom - growth enhanced, small scale growth is enhanced, small scale power increases, larger . power increases, larger sigma8 31 8 Alkistis Pourtsidou Swansea University, March 17th 2015 Models of Dark Matter coupled to Dark Energy Type 2 Type 2 Type 2 Type-2 models are classified via Type-2 models are classified via L(n, Y, Z, )=F (Y, )+f(n, ZL()n, Y, Z, )=F (Y, )+f(n, Z) Type-2 modelsWe are choose classified a sub-case via for which theWe background choose a sub-case CDM for equation which the is background CDM equation is solved to give solved to give 3 0 0 ¯ 1 0 Type 2 models. L(n, Y, Z, )=F (Y, )+f(n,3 Z¯ 1) ex: ⇢¯c =¯⇢c,0a Z ⇢¯c =¯⇢c,0a Z 0 Since Z¯ = ¯˙/a, ⇢¯ depends on the time derivative ¯˙ instead of ¯ We choose a sub-case¯ for which¯˙ the background CDM equationc ¯˙ is ¯ Type 2: Matter power spectraSince Z = /a, ⇢¯c depends on theitself time which derivative is a notable diinstead↵erence of from the Type-1 case. solved to give itself which is a notable di↵erence from the Type-1 case. 0 3 ¯ 1 ⇢¯c =¯⇢c,0a Z 0 ˙ ˙ ¯ ¯ β =0 ¯ ¯ Since10000 Z = /a, ⇢¯c depends on0 the time derivativeAlkistis Pourtsidou Swansea instead University, March of 17th 2015 Models of Dark Matter coupled to Dark Energy β =1/11 itself which is a notable di↵erence0 from the Type-1 case. Alkistis Pourtsidou Swansea University, March 17th 2015 Models of Dark Matter coupled to Dark Energy ) 3 100 Type 2: CMB temperature spectra Mpc -3 (k) (h 0 P 1 0.01 Alkistis Pourtsidou Swansea University, March 17th 2015 Models of Dark Matter coupled to Dark Energy 0.0001 0.01 1 100 -1 k (h Mpc ) Alkistis Pourtsidou Swansea University, March 17th 2015 Models of Dark Matter coupled to Dark Energy 32 Alkistis Pourtsidou Swansea University, March 17th 2015 Models of Dark Matter coupled to Dark Energy Cosmology: Type 3 TypeType 3 3 Type 3 Cosmology: Type 3 Type-3 models are classified via We consider F = Y + V ( )+ (Z) Type-3 modelsType-3 are classified models are via classified via L(n, Y, Z, )=F (Y,Z, )+f(n) 2 Type 3 models. L(n, Y, Z, )=L(n,F Y,(Y,Z, Z, )= )+FWe(fY,Z,(n consider) ex: )+WefF(n choose=) Y + V a( sub )+ case (Z) with (Z)= 0Z ¯ 2 Type 3 is special: J0 =0¯ ¯ We chooseType a sub 3 case is special with (Z)=no coupling0Z appears at the background level Type 3 Typemodels 3 is special:haveType J 30 is=0 special: TheyJ0 =0 involve pure momentum transfer fluid equations ! no coupling at the background field equations! Type 3 is special no coupling appears at the background level no coupling at the background field equations! no coupling at the background field equations!fluid equations ! ˙ We also derive the perturbed KG and perturbed CDM equations. ⇢¯c +3 ⇢¯c =0 ˙ ⇢¯˙c +3H ⇢¯c =0⇢¯c +3 ⇢¯c =0 This is a pure momentum-transfer coupling up-to linear order. H HWe also derive the perturbed KG and perturbed CDM equations. Furthermore, the energy-conservation equationType 3: remains CMBThis is uncoupled atemperature pure momentum-transfer spectra coupling up-to linear order. Type 3: MatterFurthermore, power spectraFurthermore, the energy-conservation the energy-conservation equation remains equation uncoupled remains uncoupled 1 even at the lineareven level, at the i.e. linear ⇢ level,/⇢¯ obeys i.e. uncoupled ⇢/⇢¯ obeys equation. uncoupled equation. ˙ = ✓ h˙ even at the linear level, i.e.⌘ ⇢/⇢¯ obeys⌘ uncoupled equation. c c 2 ⌘ ˙ 1 ˙ c = ✓c ˙h Type-3 is a pure momentum-transfer theory. 2✓c = ✓c S( 0) Type-3 is a pureType-3 momentum-transfer is a pure momentum-transfer theory. theory. H ✓˙ = ✓ S( ) c H c 0 γ =0 10000 0 γ0=0.15 Alkistis Pourtsidou Swansea University, March 17th 2015 Models of Dark Matter coupled to Dark Energy Alkistis Pourtsidou Swansea University, March 17th 2015 Models of Dark MatterAlkistis coupled Pourtsidou to DarkSwansea Energy University, March 17th 2015 Models of Dark Matter coupled to Dark Energy Alkistis Pourtsidou Swansea University, March 17th 2015 Models of Dark Matter coupled to Dark Energy ) Alkistis Pourtsidou Swansea University, March 17th 2015 Models of Dark Matter coupled to Dark Energy 3 100 Mpc -3 (k) (h 0 P 1 0.01 0.0001 0.01 1 100 -1 k (h Mpc ) Parameterise these mixed models - extend the PPF formalism [Hu 08, Skordis 08] [Skordis, Pourtsidou,33 EC 2015] Alkistis Pourtsidou Swansea University, March 17th 2015 Models of Dark Matter coupled to Dark Energy Alkistis Pourtsidou Swansea University, March 17th 2015 Models of Dark Matter coupled to Dark Energy Summary 1.Depending on your faith in string landscape approach we have a solution to CC problem. If not, its solution remains to be determined. 2. Sequestering cancels CC at all orders - impact of gravity loops ? 3. Fab Four - provides a way of living with a large changing cosmological constant ! Realistic models ? 4. Quintessence type approaches require light scalars which bring with them fifth force constraints that need satisfying. 5. Need to screen this which leads to models such as axions, chameleons,non- canonical kinetic terms etc.. -- these have their own issues. 6. Alternatively could consider modified gravity such as massive gravity but this brings with it constraints. 7. Increased interest in coupled DE-DM models which can be analysed by PPF formalism and can include new couplings such as scalar field to velocity components. 34 Extra slides if time permits 35 Designer f (R) or f(G) models [Hu and Sawicki (2007), ...] Construct a model to satisfy observational requirements: 1. Mimic LCDM at high z as suggested by CMB 2. Accelerate univ at low z 3. Include enough dof to allow for variety of low z phenomena 4. Include phenom of LCDM as limiting case. Effective chameleon mechanism 36 What should we do to help determine the nature of DE ? 1. We need to define properly theoretically predicted observables, or determine optimum ways to parameterise consistency tests (i.e. how should we parameterise w(z)?) 2. Need to start including dynamical dark energy, interacting dark matter- dark energy and modified gravity models in large scale simulations - [Wyman et al 2013, Li et al 2013 Puchwein et al 2013, Jennings et al 2012, Barreira et al 2012, Brax et al 2013]. 3. Include the gastrophysics + star formation especially when considering baryonic effects in the non-linear regimes - `mud wrestling’. 4. On the theoretical side, develop models that go beyond illustrative toy models. Extend Quintessential Axion models. Are there examples of actual Landscape predictions? De Sitter vaccua in string theory is non trivial. 5. Recently massive gravity and galileon models have been developed which have been shown to be free of ghosts. What are their self- acceleration and consistency properties? 37 6. Will we be able to reconstruct the underlying Quintessence potential from observation? 7. Will we ever be able to determine whether w≠-1 ? 8. Look for alternatives, perhaps we can shield the CC from affecting the dynamics through self tuning-- The Fab Four 9. Given the complexity (baroque nature ?) of some of the models compared to that of say Λ, should we be using Bayesian model selection criterion to help determine the relevance of any one model. Things are getting very exciting with DES beginning to take data and future Euclid missions, LSST, as well as proposed giant telescopes, GMT, ELT, SKA - travelling in new directions ! 38 NonPC Caveats on PCs • Principal component decomposition of w(z) shows many components can in principle be measured (Euclid, LSST, WFirst) What’s the best• wayYet even to paramemodels liketerise Albrecht-Skordis the DE eqn (oscillating of state w) ?are still dominated by first component - average w or pivot - plus 1-2 weaker Important •for Should surveys a multidimensional like DES, parameterizationEUCLID, LSST be the basis for optimizing an experiment? 1. Principal components - [Mortonson,Hu w(z) w (z)= ↵ e (z) and Huterer b i i i 2011] X w baseline eos b e Fischer matrix eig modes i 2. w(z) 39 2 us to consider a completely free function w(z). As this anisotropy power spectrum can provide information on corresponds to an infinite number of new degrees of free- particular combinations of the cosmological parameters. dom, we have to simplify the problem. We use a physi- For instance the relative height of the Doppler peaks de- cally motivated parametrisation where w = p/ρ is defined pends on the baryon density and the scalar spectral in- by its present value, w0,itsvalueathighredshift,wm, dex. In order to break such degeneracies it is necessary to the value of the scale factor where w changes between add external information. Since our goal is to constrain these two values, at and the width of the transition, ∆. the properties of the dark energy, in a flat geometry the Namely: main limitation comes from the geometric degeneracy be- tween w0,thedarkenergydensityΩDE and the Hubble w(a)=w0 +(wm w0)Γ(a, at, ∆)(1) − parameter h.Thisdegeneracycanbebrokenbyassum- ing an HST prior on the value of h [30] or/and combining where Γ,thetransitionfunction,hasthelimitsΓ(a = the CMB with other data sets such as the matter power 0) = 1 and Γ(a =1)=0andvariessmoothlybe- spectrum measurements from the 2dF galaxy survey [31] tween these two limits in a way that depends on the or the SN-Ia data. In our analysis we use the “gold” sub- two parameters a and ∆ (see figure 1). Such a choice t set of the recent compilation of supernova data of [25] has been shown to allow adequate treatment of generic in addition to the WMAP TT and TE spectra. An im- quintessence and to avoid the biasing problems inherent portant point which we want to stress here is that CMB in assuming that w is constant. Two choices for Γ have and SN data can be treated at a fundamental level with- been given in the literature [23, 26] as discussed in Ap- out any prior assumption on the underlying cosmological pendix A. Here we use the form advocated in [26]. model. For instance, this is the case for the matter power spectrum data from galaxy surveys which implicitely as- w(a) sumes a ΛCDM model when passing from redshift space w to real space. For this reason we add the 2dFGRS large m scale structure data only in order to check the stability of our results. We also remark that the use of secondary observables such as the age of the Universe, the size of allows for tracker like behaviour w 0 the sound horizon at the decoupling, the clustering am- although very tight bounds plitude σ8 (as quoted by the WMAP-team) or the growth D factor of matter density perturbations should not be used emerging from Planck on allowed without thought to infer constraints on the dark energy since their quoted value is usually derived by implicitly a assuming a ΛCDM cosmology. This leads to biased re- density of early dark energy 1 at Small z Large z sults since these observables depend on the nature of the dark energy [28, 32, 33]. In principle CMB constraints can be easily added by using the position of the Doppler FIG. 1: Schematic plot of the equation of state paratrisation Eq. (1). peaks, which provide an estimate of the angular diameter distance to the last scattering surface. But it is a well 10 known fact that pre-recombination effects can shift the Using this general prescription has a profound advan- 2 peaks from their true geometrical position [32]. tage in attempts to detect dark energy dynamics since,could be performed numerically. 3. w(ΩDE) unlike simpler parametrisations based on only one orFinally,The it is Integrated interesting Sachs-Wolfe to make the (ISW) connection effect be- also induces two1.5 variables, it can accurately describe both slowly andtween ouran additional parametrisation shift and in the the position dark energy of the ‘flow first pa- peak, in a rapidly varying equation of states [27]. Detecting darkrameter’way that is strongly dependent on the evolution of the 1 dark energy equation of state and is generally larger than energy dynamics and distinguishing it fromSUGRA a cosmolog- 1+we in ΛCDM modelsF [28].2 Therefore, a consistent(37) dark en- ical constant is difficult and is clear only when there are ⌘ ⌦e 0.5 ergy data analysis of the CMB indeed requires the com- rapid,(z=0) late-time, changes in w [28], which then needs a ! that was introduced in [45, 46] (see also [47]). Here, formalismw capable of describing such rapid transitions. putation of the CMB power spectrum (and TE cross- = Vcorrelation). /V ,whereV ( ) is the scalar field dark energy 0 In order to compute the CMB power spectra, we usepotential a and V is the derivative of V with respect to modified version of the CMBfast Boltzmann solver [29]. . By consideringEach of our general models dark is energy then models characterised where the by the dark This is a non-trivial step in the case where Γ changes energy parameters W DE =(w0,wm,at, ∆)andthecos- −0.5 field either accelerates or decelerates down its potential2 good for dynamical DE such as rapidly (such as in our best-fit model!). In1EXP fact usingtoward a mological its minimum parameters (dubbed ‘thawing’W C =( orΩ ‘freezing’DE, Ωbh field,h,nS, τ,As), which are the dark energy density, the baryon density, numerical−1 method that is not able to track rapid transi-evolution [48]), the authors of [45] were able to demon- −2 −1.8 −1.6 −1.4 −1.2 −1 −0.8 the Hubble parameter, the scalar spectral index, the op- Quintessence as long as monotonic tions can lead to errors significantlyw(z=0) larger than the errorstrate that F remains nearly conserved until quite recent bars on the data, of order 5%, and consequently leadtimes, ticalz 1 depth2, after and which the overall dark energy amplitude finally of begins the fluctuations FIG. 5. The 2D 68% (dark shading) and 95% (light shad- ⇡ evolution. to completely wrong results. Our tests are describedto in takerespectively. over. This is Wedespite thereforewe, ⌦e endand upall with being ten dy- parameters ing) marginalised contours in the we z=0–we0 z=0 plane for namical.which This can constant be varied nature independently. of the flow parameter is a thedetail 1EXP in and Appendix SUGRA B quintessence and C. | models| superimposed direct consequence of the fact that the dark energy field 2 uponSeveral the corresponding degeneracies contours amongst of the the dark cosmological energy clock param- There is a remaining degeneracy in nS, τ and Ωbh , 40 does not exist in a vacuum: instead it has been influenced parametrisation.eters prevent us from accurately constraining cosmologi- which allows the models to reach unphysically high values by the long periods of radiation and matter dominated cal models using CMB data only. Specific features of theepochsof prior the to baryon the current density day. and the reionisation optical depth. ever w = 0. Many scalar field models of dark energy If the parameter is a constant, (as is the case for ex- e ponential potentials) or remains approximately constant, possess scaling solutions on which the dark energy equa- 2 then we have we = 1+F ⌦e, which looks very much tion of state we ‘tracks’ that of the dominant background like our dark energy clock parametrisation with w0 = 1 component (see e.g. [29, 30]). In such models, mono- 2 and w1 = F . This indicates that, so long as F re- tonicity of ⌦e would be spoiled as we evolves through the radiation and matter dominated eras and transitions mains approximately constant throughout the long radi- ation and matter dominated eras, then the behaviour of towards we 1 today. In this paper, we have used our parametrisation⇠ to probe the dynamics of dark en- a wide range of scalar field dark models should be well ergy at low redshifts, z =0 1.75. It is presumably safe captured by we(⌦e)=w0 + w1⌦e. to assume that ⌦e remains monotonic over this redshift range, since to break monotonicity between z = 0 and ACKNOWLEDGEMENTS z 2 would require a rapidly varying we, which is not favoured⇠ by existing analysis [14, 23]. Hence, in our fit- The authors would like to thank Arman Shafieloo for ting to data we imposed the hard prior we(z) < 0 for z =0 1.75. inspiring discussions during the inception of this work. To probe the high redshift behaviour of dark energy We would also like to thank Adam Moss, Mattia Fornasa, this prior would need to be removed, since it would be un- Anastasios Avgoustidis and especially Ren´ee Hlozek for useful discussions. ERMT is supported by the University realistic to say with any sort of certainty that we(z) < 0 throughout the entire cosmic history. Hence, to accom- of Nottingham, EJC acknowledges The Royal Society, Leverhulme and STFC for financial support and AP and modate high redshift data the non–monotonicity of ⌦e would need to be accounted for. This could be achieved CS were funded by Royal Society University Research Fellowships. ERMT would like to thank Sir Bradley Wig- by piece–wise parametrising we in regions of monotonic gins for winning the 2012 Tour de France. ⌦e. For example, if radiation can be neglected, these re- gions are defined by the roots of the polynomial we(⌦e)= ˜ n1 w˜nUn(⌦e) = 0. If we(⌦e)=w0 + w1⌦e and w1 > 0, Appendix A: The Chebyshev Polynomials of the then there would be two distinct regions: ⌦˙ > 0 for P e Second Kind ⌦ < w /w (I) and ⌦˙ < 0 for ⌦ > w /w (II). One e 0 1 e e 0 1 would then write The Chebyshev polynomials of the second kind are de- (I) (I) w0 fined by the recurrence relation w0 + w1 ⌦e , ⌦e < w we(⌦e)= 1 (36) (II) (II) w0 U0(x)=1,U1(x)=2x, (w0 + w1 ⌦e , ⌦e > w1 Un+1(x)=2xUn(x) Un 1(x) , (A1) and so there would be four free parameters in total. To and obey the following orthogonality condition accommodate CMB data, radiation can not be neglected, 1 but such regions of monotonicity can still be defined: one 2 ⇡ Un(x)Um(x) 1 x dx = nm . (A2) would need to compute the roots of Eq. (3) exactly, which 1 2 Z p 4 B. EDE 1 4 The first parametrization that we investigate numerically (EDE1) is the one proposed in [6] and tested in refs.[7–9]. The dependence of the dark energy fraction, ⌦de,onthescaleparameterB. EDE 1 a is given by (0) 3w ⌦ ⌦ (1 a 0 ) The first parametrization that we investigatede numericallye (EDE1) is the one proposed3w0 in [6] and tested in refs.[7–9]. ⌦de(a)= + ⌦e(1 a ). (5) 0 0 3w0 The dependence of the dark energy fraction, ⌦⌦de,onthescaleparameter+ ⌦ma a is given by (0) ⌦ ⌦ (1 a 3w⌦0 ) ⌦ Eq. (5) uses three parameters, the present matterde e fraction m0,theearlydarkenergyfraction3w0 e and the present ⌦de(a)= + ⌦e(1 a ). (5) dark energy equation of state w0, with ⌦de0 =1⌦0 +⌦⌦m00.Foranygivenfunctiona3w0 ⌦de(a) the scale dependent equation de m of state w(a) obtains as a simple derivative [33] Eq. (5) uses three parameters, the present matter fraction ⌦m0,theearlydarkenergyfraction⌦e and the present dark energy equation of state w0, with ⌦de0 =1 1⌦m0.Foranygivenfunctiond ln ⌦de aeq ⌦de(a) the scale dependent equation w(a)= + , (6) of state w(a) obtains as a simple derivative 3[1 [33] ⌦ (a)] d ln a 3(a + a ) de eq 1 d ln ⌦de aeq where aeq 3200 is the scale factorw at(a matter-radiation)= equality,+ while the energy, density is given by (6) ⇠ 3[1 ⌦ (a)] d ln a 3(a + a ) de eq ⇢ ⌦ ⌦ 1 a 1 a 3200 de0 de de eq where eq is the scale factor⇢de(a at)= matter-radiation3 equality, 1+ while the energy. density is given by (7) ⇠ a ⌦de0 ⌦de0 1 a 1+aeq ⇣ ⌘ ⇢de0 ⌦de ⌦de 1 aeq 1 ⇢de(a)= 3 1+ . (7) a ⌦de0 ⌦de0 1 a 1+aeq C. EDE⇣ 2 ⌘ 4 The second parametrization (EDE2),How that early we propose isC.B. early hereEDE EDE dark for 1 2 the energy? first time, [Pettorino,Amendola reads as follows: and Wetterich 2013]