Dark Energy and the Formation of the Large Scale Structure of the Universe

124 Pages·2016·3.248 MB·English

by  Jérôme Gleyzes (auth.)

Springer Theses
Recognizing Outstanding Ph.D. Research
Jérôme Gleyzes

Dark Energy and
the Formation of
the Large Scale
Structure of the
Universe
Springer Theses
Recognizing Outstanding Ph.D. Research
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é ô
J r me Gleyzes
Dark Energy and the
Formation of the Large Scale
Structure of the Universe
Doctoral Thesis accepted by
the Institute for Theoretical Physics-CEA Saclay, France
123
Author Supervisor
Dr. Jérôme Gleyzes Dr. Filippo Vernizzi
Jet Propulsion Laboratory CEA, IPhT
Pasadena,CA Gif-sur-Yvette
USA France
ISSN 2190-5053 ISSN 2190-5061 (electronic)
SpringerTheses
ISBN978-3-319-41209-2 ISBN978-3-319-41210-8 (eBook)
DOI 10.1007/978-3-319-41210-8
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’
Supervisor s Foreword
In the last 30 years, cosmology has witnessed much progress mostly spurred by a
positive interplay between theoretical developments and observational discoveries.
Thanks to high-precision data such as the cosmic microwave background aniso-
tropies, we have been able to reconstruct the evolution of the universe and of the
structuresthatweobservetodaywithanexquisiteaccuracy.Forinstance,weknow
that large-scale structures formed by the gravitational collapse of dark matter
aroundsomesmallinitialinhomogeneities.Accordingtoourcurrentunderstanding,
the initial seeds of these inhomogeneities were generated during inflation, an early
universephasecharacterisedbyanacceleratedexpansion,triggeredbythevacuum
energy of a scalar field. The quantum fluctuations of this field, converted into
density perturbations, are imprinted in the observable sky. By studying their sta-
tistical distribution we have been able to confirm inflationary predictions. Despite
these confirmations, inflation remains a paradigm awaiting for a convincing proof
and lacking robust connections with know high energy physics.
Even more mysterious, the current accelerated expansion of the universe, dis-
covered in 1998, has triggered enormous interest in both the communities of
observational cosmologists and theoretical physicists. The simplest explanation, a
cosmological constant is consistent with observations. However, given the theo-
retical difficulties traditionally associated to this possibility, most of the efforts
of the scientific community are now devoted to rule it out. Alternatively, the
acceleration may be due to some dynamical component called dark energy or to
some modification in the laws of gravity on very large scales. Many models have
been proposed, each of them leading to specific effects on the evolution of struc-
tures formation. As for inflation, the knowledge of the statistical properties of the
large-scalestructuresandtheirevolutionmaybecriticaltohelptobetterunderstand
theoriginofthecurrentacceleration.Forthisreason,themajorscienceagenciesare
currently planning large field cosmic surveys. The main scientific driver of these
ground based and space telescopes is a precise determination of the statistical
properties of the cosmic fields and their evolution, with the primary goal of con-
straining dark energy and the origin of cosmological perturbations.
v
vi Supervisor’sForeword
JérômeGleyzesstartedtoworkonhisPh.D.thesisinthisscientificcontext.He
first focused his research on bridging theoretical models of dark energy with
observations. It should be emphasised that the number and variety of proposed
modelsofdarkenergyandmodifiedgravityisratherimpressive,whichrepresentsa
challenge for future observations. Jérôme developed an approach to characterise
most models in a unified way, in terms of a few number of parameters corre-
spondingtoparticularobservationaleffectsoncosmologicalscales.Thisisbasedon
the construction of a general theory of cosmological perturbations around a cos-
mological background in terms of all possible Lagrangian operators satisfying
certain symmetries, dictated by the class of models under consideration. This
approachhasrapidlybecomepopularinthescientificcommunityunderthenameof
effective fieldtheory ofdark energy andwill likelyplay animportant role inyears
tocomeinattemptstoconstraindeviationsfromgeneralrelativityoncosmological
scales.
Indevelopingthisapproach,toavoidmodelswithinstabilitiesJérômerestricted
to the quadratic operators leading at most to two derivatives in the equations of
motion for the propagating degrees of freedom. However, he discovered that the
quadraticoperatorssatisfyingthese properties were onemore than those needed to
describe the so-called Horndeski theories. This came as a surprise, because such
theories, developed by Horndeski in the seventies and recently rediscovered, were
long believed to be the most general ones being free from dangerous instabilities.
BycompletingtheLagrangianofthequadraticoperatorsatthefullnon-linearlevel,
he was able to construct theories beyond Horndeski. These theories allow higher
derivativesintheequationsofmotions.However,duetotheirdegeneracytheyonly
contain propagating degrees offreedom whose order of derivatives is never higher
thantwo,asrequiredforahealthytheorywithoutinstabilities.Thisdiscoveryledto
a very rich activity in the literature, with many studies of the phenomenological
consequences of theories beyond Horndeski and several theoretical developments
on their extensions.
In 2014, triggered by the exciting—but unfortunately incorrect—conclusions
that the BICEP2 telescope had observed primordial gravitational waves from
inflation, Jérôme turned to the study of inflationary predictions. As I mentioned
above, current observations confirm inflation. However, incontestable evidence
could only come from observing primordial tensor modes. In this context, Jérôme
demonstratedthatthepredictionsforthegravitationalwavespectrumfrominflation
arecompletelyrobust.Inparticular,despitesomeclaimsintheliterature,heshowed
thatitisnotpossible toalterthestandard predictionsofinflationbymodifyingthe
speedofpropagationoftensors.Incontrasttowhathappensforscalarfluctuations,
a scale-invariant spectrum of tensor fluctuations can only arise in inflation. Thus,
the measurement of the gravitational wave amplitude would unambiguously
determine the energy scale of inflation.
Finally, the last part of the thesis is devoted to the so-called consistency rela-
tions, relations between correlation functions of the cosmic fields (e.g. the dark
matter density contrast or the galaxy number density), valid in the limit in which
one of the wavelength modes is much longer than the others. These relations are
Supervisor’sForeword vii
non-perturbative for the short-scale modes; in other words they automatically
incorporate short-scale baryonic effects and the bias between galaxies and dark
matter. For this reason, they can be employed as a test of standard cosmology.
Moreover, Jérôme showed that since they are based on the equivalence principle,
theirbreakingcanbeusedtotestthepresenceofafifthforce,whicharisesinsome
modified gravity models.
In conclusion, Jérôme’s thesis covers many aspects of modern cosmology. It
provides an innovative and pedagogical introduction to each of them and contains
cutting-edge results, rare tobe found in aunique researchmanuscript. It should be
clearthatJérôme’sresearchhasalreadymadealargeimpactinthecommunityand
will surely continue to do so in the future.
Gif-sur-Yvette, France Dr. Filippo Vernizzi
April 2016
Acknowledgements
This Ph.D. has been a great adventure, where I interacted with a number of
excellent people, for which I am extremely grateful. A special mention to the
students at ICTP, like Marko and Gabriele, whom I visited numerous times. To
Michele, I say thank you for sharing the work, writing notes, and being there to
compare our codes.
Ihavelearntalotfrommanygreatscientists.Inparticular,Iwouldliketothank
ClaudiadeRham,AndrewTolley,Justin KhouryandMarkTroddenfor thetime I
had with them where our interactions were very fruitful. I would also like to give
many thanks to Pedro Ferreira and Tessa Baker. My visits to Oxford were always
enriching.
Finally, I would like to give special thanks to the people that played a crucial
roleduringmyPh.D.andhaveshapedtheresearcherIamtoday:PaoloCreminelli,
David Langlois, Federico Piazza. The most important of them is Filippo Vernizzi,
myadvisor.Thanksforalwaysbeingtheretoanswermyquestions,fortreatingme
like an equal, for giving me so many opportunities to travel and present our work,
and for guiding me through many aspects of the world of physicists.
ix
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 The Homogeneous Universe. . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 The Friedmann-Lemaître-Robertson-Walker Metric . . . . . . 2
1.1.2 Comoving Distance and Redshift . . . . . . . . . . . . . . . . . . 3
1.1.3 The Friedmann Equations. . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.4 Observations and the Discovery of Dark Energy. . . . . . . . 5
1.2 The Large Scale Structure of the Universe . . . . . . . . . . . . . . . . . 8
1.2.1 Growth of Perturbation in ΛCDM. . . . . . . . . . . . . . . . . . 10
1.2.2 Galaxy Surveys. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2.3 Weak Lensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3 This Thesis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2 The Effective Field Theory of Dark Energy. . . . . . . . . . . . . . . . . . . 21
2.1 The Unitary Gauge Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2 ADM Formalism and the Effective Field Theory
of Dark Energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2.1 Background Evolution. . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2.2 The Quadratic Action . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3 Going from Models to the EFT of DE. . . . . . . . . . . . . . . . . . . . 32
2.4 Stability and Theoretical Consistency. . . . . . . . . . . . . . . . . . . . . 35
2.5 Evolution of Cosmological Perturbations . . . . . . . . . . . . . . . . . . 37
2.5.1 Vector Sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.5.2 Tensor Sector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.5.3 Scalar Sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Appendix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3 Beyond Horndeski. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.1 Horndeski Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.2 General Considerations on Higher Order Derivatives . . . . . . . . . . 55
xi