ΛCDM Model and Hubble Tension

Jaeok Yi1

1Department of Physics, KAIST, Daejeon 34141, Korea(Dated: November 1, 2019)

ΛCDM model is the simplest model for our universe. It includes ordinary matter, cold dark matter,and dark energy (cosmological constant). Due to its simplicity and usefulness, it is considered as thestandard cosmological model. However, there are some limitations. Especially, the new measurementof Hubble constant shows deviation with old one. This Hubble tension might suggest beyond theΛCDM model.

INTRODUCTION

Many physicists believe that ordinary matter accountsfor a very tiny amount of energy of our universe. Un-detected dark matter and dark energy are believed toaccount for the most energy of the universe. They havenot been detected yet, but there is sufficiently much evi-dence for them.

FIG. 1. Current Energy Composition of the Universe [1]

Since we do not know much about the dark sector, wecannot build a model for our universe perfectly. How-ever, using Occum’s razor, we can think of the simplestmodel for our universe incorporating the dark sector. Itis known as the ΛCDM model. It provides good expla-nations for some known properties of our universe.

But it has some limitations such as vacuum energyproblem and small scale crisis. Recently, experimentsconfirmed that the values of Hubble constant measuredby Planck Mission and Hubble Space Telescope show dif-ference. This difference known as Hubble tension can beevidence for beyond the ΛCDM model.

In this note, I will review the ΛCDM model. Also, theintroduction for the ways measuring the Hubble constantwill follow. Finally, I will discuss Hubble tension and theneed for beyond the ΛCDM model.

ΛCDM MODEL

ΛCDM model, the abbreviation of Λ cold dark mattermodel, is the simplest model for our universe. ΛCDMmodel consist of three components. One is dark energy

described by the cosmological constant Λ. Another com-ponent is cold dark matter(CDM). The other one is or-dinary baryonic matter. In this section, Λ and CDMwill be explained and discussions of ΛCDM model willbe followed.

Cosmological Constant, Λ

It is well known that Einstein wanted a static universewhen he developed his field equation at first. So he wasvery disappointed when it did not allow the static uni-verse. To support the static universe, he introduced acosmological constant Λ and derived this result.

Rµν −1

2Rgµν + Λgµν =

8πG

c4Tµν

Later, Hubble discovered that the universe is expand-ing. It means that our universe is not static. So Einsteinwithdrew his cosmological constant and he called it as his”biggest blunder.” But as cosmological data are much in-creased and accurate, scientists discovered that our uni-verse is expanding at an accelerating rate. To explainthis, a kind of negative pressure is needed. The notion ofdark energy was suggested and scientists started reusingthe cosmological constant Λ.

The cosmological constant is the simplest way to de-scribe dark energy. Also, quantum field theories sug-gest its origin. They predict vacuum fluctuations whichwould give this kind of energy. Although the expectedvalue from quantum field theory is much larger than ex-perimental prediction, the current cosmology accepts Λbecause of its simplicity.

Cold Dark Matter

Dark matter is undetected mass in the universe. Al-though it have not been detected, there is much evidenceof its existence. Galaxy rotation curve and gravitationallensing are evidence for dark matter. Using these, we canassume following properties.

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• Non-baryonic : It consists of matter other thanbaryons (and electrons). If not, it interacts withlight.

• Dissipationless : It cannot cool by radiating pro-cess.

• Collisionless : It interact with each other and otherparticles only through gravity and possibly theweak force.

Also we can assume coldness of particle. Cold darkmatter means that its velocity is far less than the speedof light. If dark matter is cold, then small objects col-lapse under their gravity and merge into more massiveones, like galaxies. If dark matter is hot, then the cos-mological objects like our galaxy should be formed byfragmentation. Since cosmic microwave background ra-diation is highly uniform, hot dark matter cannot formgalaxies.

Therefore most cosmologists favor the cold dark mattertheory. It explains how galaxies and cosmological objectsare formed from a smooth initial state. Of course somecosmologists suggest warm dark matter. But to explainour universe in the simplest way, cold dark matter ispreferred.

Parameters and Successes of ΛCDM

Using the cosmological constant Λ and cold dark mat-ter, we can build a model for our universe. Assumptionfor ΛCDM model is listed below. [2]

• Physics is the same throughout the observable uni-verse.

• General Relativity is an adequate description ofgravity.

• On large scales the Universe is statistically thesame everywhere (initially an assumption, or “prin-ciple,” but now strongly implied by the nearisotropy of the CMB).

• The Universe was once much hotter and denser andhas been expanding since early times.

• There are five basic cosmological constituents:

– Dark energy that behaves just like the energydensity of the vacuum.

– Dark matter that is pressureless (for the pur-poses of forming structure), stable and inter-acts with normal matter only gravitationally

– Regular atomic matter that behaves just likeit does on Earth.

– The photons we observe as the CMB.

– Neutrinos that are almost massless (againfor structure formation) and stream like non-interacting, relativistic particles at the time ofrecombination.

• The curvature of space is very small.

• Variations in density were laid down everywhereat early times, and are Gaussian, adiabatic, andnearly scale invariant (i.e., proportionally in all con-stituents and with similar amplitudes as a functionof scale) as predicted by inflation.

• The observable Universe has “trivial” topology(i.e., like R2). In particular it is not periodic ormultiply connected.

In these assumptions, we need at least six parameters.One choice of these parameters is like below.

• Density of baryons Ωb

• Density of cold dark matter Ωc

• Amplitude of a power-law spectrum of adiabaticperturbations As

• Scalar spectral index of a power-law spectrum ofadiabatic perturbations ns

• Angular scale of acoustic oscillations θ∗

• Optical depth to Thomson scattering from reion-ization τ

Using only these six parameters, we can describe ouruniverse. Other parameters can be derived by these.ΛCDM model can explain these phenomena from sim-ple foundation.

• The existence and structure of the cosmic mi-crowave background

• The large-scale structure in the distribution ofgalaxies

• The abundances of hydrogen (including deu-terium), helium, and lithium

• The accelerating expansion of the universe observedin the light from distant galaxies and supernovae

It is considered as the standard cosmological model,due to its simplicity and usefulness. Of course ΛCDMmodel is a kind of assumption. There are some limita-tions which cannot be explained by ΛCDM model.

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HUBBLE CONSTANT MEASUREMENT

Hubble detected that the farther nebulae are movingaway more quickly. The recession velocity of nebula isapproximately proportional to its distance from us. Thisis known as Hubble-Lemaıtre law.

v = Hd

H which is a ratio of the recession velocity to theproper distance is called Hubble parameter. Since ouruniverse is expanding at an accelerating rate, H is not aconstant. But in the historical sense, the current valueof H is called Hubble constant and denoted by H0.

Since H0 is needed to estimate the size and the ageof universe, it is one of the most important quantities.But it is not easy to measure since the measurement ofdistance of the cosmological object is very difficult. Pre-cise measurement is made by Planck Mission and HubbleSpace Telescope only recently. In this section, the waysof measurement will be explained and their difference willbe compared.

FIG. 2. Pictures of Planck(left) and HST(right) [3] [4]

Hubble Constant from Planck Mission

Planck was ESA’s mission to observe the first light inthe universe. [5] It was designed to image the temper-ature and polarization anisotropies of the Cosmic Back-ground Radiation Field over the whole sky, with unprece-dented sensitivity and angular resolution. By observ-ing CMB, huge amount of data related to cosmology aremade. The data of Planck are uploaded at [6].

Data from Planck constrain 6 parameters governingΛCDM model. Also using these, other cosmologicalquantities including Hubble constant are calculated. In2018, Planck releases final results. The value they releaseis following.

H0 = 67.66 ± 0.42 km/s/Mpc

Hubble Constant from HST

Although Planck wins a tremendous success, the wayto calculate of H0 by Planck is not intuitive. The directway to measure H0 is measuring v and d for cosmolog-ical objects and calculating H0 = v/d. Hubble SpaceTelescope measure H0 using this natural way.

The recession velocity of cosmological object can bemeasured by measuring the red shift z. But it is not easyto measure the distance of cosmological object. To do it,HST uses Type Ia supernovae and cepheid variables. [8]

Type Ia supernovae are some of the best way to de-termine the extragalactic distance. They show a charac-teristic light curve. In other words, their luminosity asa function of time after the explosion is similar to eachother. Since we know their luminosity, we can measuretheir distance by measuring their apparent magnitude.Also, the cepheid variables show a characteristic period-luminosity relation.

FIG. 3. Characteristic curve of Type Ia supernovae(left) andP-L relation of cepheid variables(right) [9] [10]

Therefore HST can measure the distance of galaxyhosting SNE Ia and cepheid variables. Using thismethod, HST release its result for H0. [11]

H0 = 74.03 ± 1.42 km/s/Mpc

Hubble Tension and ΛCDM Model

There exists the 4.4σ difference between local measure-ments of H0 by HST and the value predicted from Planck+ ΛCDM. It means that ΛCDM model is not sufficientto explain whole history of universe. HST measured thedata from late universe and Planck did from early uni-verse.

This difference is called Hubble tension. It seems thatthere should be new physics that resolve Hubble tensionbetween early and late universe. Possible physics causesfor a 2–4% change in H0 include time dependent dark en-ergy or nonzero curvature, while a larger 5–8% differencemay come from dark matter interaction, early dark en-ergy or additional relativistic particles. Anyway, it seemstrue that ΛCDM model is not perfect.

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FIG. 4. 4.4σ difference between measurements of H0 fromPlanck and the value predicted from Planck + ΛCDM. [11]

SUMMARY AND CONCLUSION

In this note, we review the ΛCDM model and Hub-ble tension. The values of Hubble constant from Planckand HST show a significant difference. There are someways to explain it, but we do not know which is correct.But one thing is clear, the universe is not simple as wethought.

[1] Vennin, Vincent. (2014). Cosmological Inflation: Theo-retical Aspects and Observational Constraints.

[2] arXiv:1807.06205[3] http://www.esa.int/ESA_Multimedia/Images/2007/

01/Front_view_of_the_Planck_satellite

[4] https://catalog.archives.gov/OpaAPI/media/

23486741/content/stillpix/255-sts/STS125/STS125_

ESC_JPG/255-STS-s125e011848.jpg

[5] arXiv:astro-ph/0604069[6] http://pla.esac.esa.int/pla/#home

[7] arXiv:1807.06209[8] A. G. Riess, L. M. Macri, S. L. Hoffmann, D. Scolnic, S.

Casertano, A. V. Filippenko, B. E. Tucker, M. J. Reid, D.O. Jones, J. M. Silverman, R. Chornock, P. Challis, W.Yuan, P. J. Brown, and R. J. Foley, The AstrophysicalJournal 826, 56 (2016).

[9] http://www.outerspacecentral.com/supernova_page.

html

[10] I. Meschin, C. Gallart, A. Aparicio, S. Cassisi, and A.Rosenberg, The Astronomical Journal 137, 3619 (2009).

[11] A. G. Riess, S. Casertano, W. Yuan, L. M. Macri, andD. Scolnic, The Astrophysical Journal 876, 85 (2019).