Constraining Gravitational and Cosmological Parameters with Astrophysical Data
\departmentDepartment of Physics \degreeDoctor of Philosophy \degreemonthJune \degreeyear2008 \thesisdateMay 23, 2008
Max TegmarkAssociate Professor of Physics \supervisorAlan H. GuthVictor F. Weisskopf Professor of Physics
Thomas J. GreytakAssociate Department Head for Education
We use astrophysical data to shed light on fundamental physics by constraining parametrized theoretical cosmological and gravitational models.
Gravitational parameters are those constants that parametrize possible departures from Einstein’s general theory of relativity (GR). We develop a general framework to describe torsion in the spacetime around the Earth, and show that certain observables of the Gravity Probe B (GPB) experiment can be computed in this framework. We examine a toy model showing how a specific theory in this framework can be constrained by GPB data. We also search for viable theories of gravity where the Ricci scalar in the Lagrangian is replaced by an arbitrary function . Making use of the equivalence between such theories and scalartensor gravity, we find that models can be made consistent with solar system constraints either by giving the scalar a high mass or by exploiting the socalled Chameleon Effect. We explore observational constraints from the latetime cosmic acceleration, big bang nucleosynthesis and inflation.
Cosmology can successfully describe the evolution of our universe using six or more adjustable cosmological parameters. There is growing interest in using 3dimensional neutral hydrogen mapping with the redshifted 21 cm line as a cosmological probe. We quantify how the precision with which cosmological parameters can be measured depends on a broad range of assumptions. We present an accurate and robust method for measuring cosmological parameters that exploits the fact that the ionization power spectra are rather smooth functions that can be accurately fit by phenomenological parameters. We find that a future square kilometer array optimized for 21 cm tomography could have great potential, improving the sensitivity to spatial curvature and neutrino masses by up to two orders of magnitude, to and eV, and giving a detection of the spectral index running predicted by the simplest inflation models.
Acknowledgments
The work presented in this thesis, and my time at MIT over the last six years, has benefited from the contributions and support of many individuals. At MIT, I would like to thank my advisors, Max Tegmark and Alan H. Guth, my thesis committee, Scott A. Hughes and Erotokritos Katsavounidis, and Hong Liu, Jackie Hewitt, Iain Stewart, Miguel Morales, Serkan Cabi, Thomas Faulkner, along with Scott Morley, Joyce Berggren, Charles Suggs and Omri Schwarz. I would also like to thank my collaborators Matias Zaldarriaga, Matthew McQuinn and Oliver Zahn at the Center for Astrophysics at Harvard University, and Emory F. Bunn at University of Richmond. The particletheory and astro grads have been a constant source of support and entertainment, especially Molly Swanson, Qudsia Ejaz, Mark Hertzberg, Onur Ozcan, Dacheng Lin, Adrian Liu, and in earlier times, Judd Bowman and Ying Liu. I would like to give my special thanks to my dad Zhenzhong Mao, my wife Yi Zheng and my sister Su Jiang, for their constant support and encouragement without which I could not accomplish the course of study.
Contents:
 \thechapter Introduction

\thechapter Constraining torsion with Gravity Probe B
 4 Introduction
 5 RiemannCartan spacetime
 6 Parametrization of the Torsion and Connection
 7 Precession of a gyroscope I: fundamentals
 8 Precession of a gyroscope II: instantaneous rate
 9 Precession of a gyroscope III: moment analysis
 10 Constraining torsion parameters with Gravity Probe B
 11 Linearized Kerr solution with torsion in Weitzenböck spacetime
 12 A toy model: linear interpolation in RiemannCartan Space between GR and HayashiShirafuji Lagrangian
 13 Example: testing Einstein HayashiShirafuji theories with GPB and other solar system experiments
 14 Conclusions and Outlook
 15 Parametrization of torsion in the static, spherically and parity symmetric case
 16 Parametrization in stationary and spherically axisymmetric case
 17 Constraining torsion with solar system experiments
 18 Constraining torsion parameters with the upper bounds on the photon mass
 \thechapter Constraining Gravity as a Scalar Tensor Theory

\thechapter Constraining cosmological parameters with 21cm tomography
 24 Introduction

25 Forecasting Methods & Assumptions
 25.1 Fundamentals of 21cm cosmology
 25.2 Assumptions about and
 25.3 Assumptions about Linearity
 25.4 Assumptions about nonGaussianity
 25.5 Assumptions about reionization history and redshift range
 25.6 Assumptions about cosmological parameter space
 25.7 Assumptions about Data
 25.8 Assumptions about Residual Foregrounds

26 Results and discussion
 26.1 Varying ionization power spectrum modeling and reionization histories
 26.2 Varying
 26.3 Varying the nonGaussianity parameter
 26.4 Varying redshift ranges
 26.5 Optimal configuration: varying array layout
 26.6 Varying collecting area
 26.7 Varying observation time and system temperature
 26.8 Varying foreground cutoff scale
 27 Conclusion & outlook
 28 goodness of fit in the MID model
 \thechapter Conclusions
List of Figures:
 1 Classification of spaces (Q,R,S) and the…
 2 Cosmic time line
 3 Thermal history of the intergalactic gas…
 4 Gravity Probe B experiment
 5 Constraints on the PPN and torsion…
 6 Constraints on the EHS parameters from…
 7 Predictions for the average precession rate by…
 8 Shapiro time delay
 9 Effective potential for the Chameleon model…
 10 Solar system constraints on the Chameleon
 11 Potential for the model
 12 Constraints on the cubic model
 13 Effective potential for the polynomial model…
 14 21 cm tomography
 15 Parametrization of ionization power spectra and…
 16 Examples of array configuration changes
 17 Available pixels from upcoming…
 18 Relative 1 error for measuring…
 19 How cosmological constraints depend on and…
 20 1 error for various…
 21 1 error for various…
 22 Cartoon showing how cosmological parameter measurement…
List of Tables:
 1 PPN parameters, their significance and experimental bounds
 2 A short list of torsion theories of gravity
 3 Cosmological parameters measured from WMAP and SDSS LRG data
 4 Constraints of PPN and torsion parameters with solar system tests
 5 Summary of metric and torsion parameters…
 6 Summary of metric and torsion parameters…
 7 Summary of the predicted precession rate for GR and EHS theories
 8 Biases of EHS theories relative to GR predictions
 9 Constraints of the PPN and EHS parameters with solar system experiments
 10 Factors that affect the cosmological parameter…
 11 The dependence of cosmological constraints on…
 12 Fiducial values of ionization parameters
 13 Specifications for 21cm interferometers
 14 How cosmological constraints depend on the ionization power spectrum modeling and reionization history
 15 marginalized errors for the ionization parameters in the MID model
 16 How cosmological constraints depend on the redshift range in OPT model
 17 How cosmological constraints depend on the redshift range in MID model
 18 Optimal configuration for various 21cm interferometer arrays
 19 How cosmological constraints depend on collecting areas in the OPT model
 20 How cosmological constraints depend on collecting areas in the MID model
 21 How cosmological constraints depend on observation time in the OPT model
 22 How cosmological constraints depend on observation time in the MID model
Chapter \thechapter Introduction
Study of gravitation and cosmology has a long history, tracing back to antiquity when a number of Greek philosophers attempted to summarize and explain observations from the natural world, and has now evolved into two successful and flourishing areas. Since Einstein’s general theory of relativity (GR) was first proposed about ninety years ago, it has emerged as the handsdown most popular candidate for the laws governing gravitation. Moreover, during the past decade a cosmological concordance model, in which the cosmic matter budget consists of about 5% ordinary matter (baryons), 30% cold dark matter and 65% dark energy, has emerged in good agreement with all cosmological data, including the cosmic microwave background observations, galaxy surveys, type Ia supernovae, gravitational lensing and the Lyman forest.
Why is gravitation on an equal footing with cosmology in this thesis? This is because they are closely related subjects: gravitation is the theoretical foundation of cosmology, and cosmology can test gravity on the scale of the universe. Gravitation has influenced cosmology right from the start: the modern Big Bang cosmology began with two historical discoveries, the Hubble diagram and the Friedmann equation. As an application of GR, the latter predicted the possibility of an expanding universe. In recent years, attempts have been made to explain away the dark energy and/or dark matter by modifying GR. Socalled gravity [10, 25, 50, 52, 56, 60, 61, 63, 73, 76, 81, 185, 188, 193, 202, 227, 277], which generalizes the gravitational Lagrangian to contain a function of the curvature , can potentially explain the latetime cosmic acceleration without dark energy, or provide the inflaton field in the early universe. DGP gravity [77], named after its inventors Dvali, Gabadadze and Porrati, adopts a radical approach that assumes that a 3dimensional brane is embedded in a 5dimensional spacetime, and also claims that it can reproduce the cosmic acceleration of dark energy. The approach of Modified Newtonian Dynamics (MOND) [169], in particular the relativistic version – Bekenstein’s tensorvectorscalar (TeVeS) theory [28] – purports to explain galaxy rotation curves without dark matter.
Turning to how cosmology has influenced gravitation, the cosmological concordance model assumes that the expansion and structure formation of the universe are governed by equations derived from GR, mostly to linear order. It is therefore not a surprise that modified theories of gravity can imprint their signatures on the expansion history and the density perturbations of the universe. Recent research in this direction has undergone rapid progresses towards the socalled Parametrized PostFriedmann formalism [33, 116, 117] that can in principle use the avalanche of cosmology data to test gravity on scales up to the cosmic horizon.
In a nutshell, gravitation and cosmology are united. To test both — which is the subject of this thesis — we will generalize the standard models of both gravitation and cosmology, such that our ignorance can be parametrized by a few constants, and constrain those constants with astrophysical data.
1 Testing gravity
1.1 Was Einstein right?
Parameter  Meaning  GR value  Effect  Limits  Remarks 
Amount of spacetime  0  (i) Shapiro time delay  Cassini tracking  
curvature produced  (ii) Light deflection  VLBI  
by unit rest mass  (iii) Geodetic precession  (anticipated)  Gravity Probe B  
Amount of “nonlinearity” in the  0  (i) Perihelion shift  from helioseismology  
superposition law for gravity  (ii) Nordtvedt effect  
Preferredlocation effects  0  Earth tides  Gravimeter data  
Preferredframe  0  Orbital polarization  Lunar laser ranging  
effects  0  Solar spin precession  Alignment of Sun and ecliptic  
0  Pulsar acceleration  Pulsar statistics  
Violation of  0  –  Combined PPN bounds  
conservation  0  Binary motion  for PSR 1913+16  
of total momentum  0  Newton’s 3rd law  Lunar acceleration  
0  –  –  Not independent 
Einstein’s GR has been hailed as one of the greatest intellectual successes of human beings. This reputation is a consequence of both its elegant structure and its impressive agreement with a host of experimental tests. In Relativity, space is not merely a stage where events unfold over time as it was in Newtonian mechanics. Rather, spacetime is a unified entity which has a life of its own: it can be sufficiently curved to form an event horizon around a black hole, it can propagate ripples in “empty” space with the speed of light, and it can expand the universe with the driving force given by the matter and energy inside it.
In GR and most generalized gravity theories, spacetime is, in mathematical language, a manifold whose geometry is dictated by the metric, a tensor defined at each location. GR contains two essential ingredients that experiments can test. The first is how spacetime influences test particles (particles small enough that they do not significantly change the spacetime around them) such as photons or astrophysical objects. In the absence of nongravitational forces, test particles move along geodesics, which are generalized straight lines in GR. I term experiments that use a planet or a light ray to probe the metric around a massive gravitating object (Sun or Earth) as geometric tests of GR. Geometric tests have included the wellknown weakfield solar system tests: Mercury’s perihelion shift, light bending, gravitational redshift, Shapiro time delay and lunar laserranging. An ongoing satellite experiment, Gravity Probe B, that measures the spin precession of freefalling gyroscopes, falls into this category too, since the spin precession is a response to the spacetime metric that arises from both the mass (geodetic precession) and the rotation (framedragging effect) of Earth. Additionally, the LAGEOS [62] experiment has directly confirmed the framedragging effect on the orbits of test particles around the rotating Earth.
In the weakfield regime, there exists a mature formalism, the socalled Parametrized PostNewtonian (PPN) formalism, that parametrizes departures from GR in terms of a few constants under certain reasonable assumptions. Thus the success of GR in this regime can be fully described by the observational constraints on these PPN parameters near their GR predictions, as summarized in Table 1.
The second part of GR describes how matter, in the form of masses, pressures and sometimes shears, curves the spacetime around them. Specifically the metric and matter are related by the Einstein field equation (EFE). I term any experiments that test EFE dynamical tests. The linearized EFE predicts the existence of gravitational waves, i.e. propagation of tensorial gravitational perturbations. The upcoming experiments LIGO and LISA are direct probes of gravitational waves from black holes or the primordial universe. An indirect probe is the observation of the damping of binary orbits due to gravitational radiation reaction in the binary pulsars. Upcoming observations of strongfield binary compact objects and blackhole accretion will be exquisite dynamical tests of GR. In addition, as mentioned above, cosmology can test the EFE, since the cosmic expansion and structure formation are determined by the zeroth and linear order EFE.
1.2 Generalizing GR
As mentioned, since GR is the foundation of both modern gravitation and cosmology theory, testing GR is of central interest in the science community. Two popular approaches have been taken to test GR in a broader framework. The first approach is to generalize GR in a modelindependent fashion by making a few assumptions that are valid in a certain limit, and parametrize any possible extensions of GR by a few constants. For example, the PPN formalism [246, 263, 264] parametrizes theories of gravity in the weakfield limit with 10 PPN parameters (see Table 1) that can be constrained by solarsystem test data. The developing Parametrized PostFriedmann (PPF) formalism, as a second example, parametrizes the cosmology of modified gravity theories to linear order in cosmic density perturbations and may end up with a few PPF parameters too, that may be constrained by cosmological experiments in the future. The second approach to testing GR is to follow the debate strategy: if we can rule out all modified theories of gravity that we can think of, then GR becomes more trustworthy. Arguably the most beautiful aspect of GR is that it geometrizes gravitation. Consequently, there are at least three general methods that can generalize GR, corresponding to different geometries.
The first method is to introduce extra dynamical degrees of freedom in the same geometry as GR. The geometry where GR is defined is the socalled Riemann spacetime, that is completely specified by the metric , a tensor at each spacetime position. In the Riemann spacetime, a freefalling particle moves along a covariantly “constant” velocity curve, in the sense that the 4D velocity vector has vanishing covariant derivative (), because the change in the absolute differentiation () is compensated for by a term involving the socalled connection that characterizes a curved manifold and defines the spacetime curvature . The connection and curvature are not free in the Riemann spacetime — they are defined in terms of the metric and its 1st and/or 2nd derivatives. The dynamics of Einstein’s GR is given by the simple action, , from which EFE is derived. Here , where is Newton’s gravitational constant and is the speed of light. The factor is inserted so that is covariant (i.e. unchanged under arbitrary coordinate transformations). The outstanding simplicity of GR is that it contains no free parameters, given that is fixed by the inversesquare law in the Newtonian limit. To generalize GR, however, one trades off the simplicity for the generalization. For example, one can take the action to contain, in principle, arbitrary functions of the curvature, i.e. — which defines socalled gravity. A new scalar field with arbitrary potential and couplings to the metric can also be introduced into the action — this is socalled scalartensor gravity. In fact, gravity is equivalent to a special class of scalartensor gravity theories. Additionally, the action can include even more fields (vector fields plus scalar fields), as in the socalled TeVeS (standing for “tensorvectorscalar”) gravity, a relativistic version of MOND.
The second method to generalize GR is to generalize the geometry such that the emergent degrees of freedom in the spacetime manifold are dynamic variables. The simplest extension to Riemann spacetime is the socalled RiemannCartan spacetime with nonzero torsion. In a nutshell, torsion is the antisymmetric part of the connection mentioned earlier – in Riemann spacetime the connection is constrained to be symmetric, so allowing for nonzero torsion relaxes this constraint. The geometry of RiemannCartan spacetime is pinned down by the metric and torsion – the socalled torsion theory is established in terms of these two pieces. Just as Riemann spacetime is a special case of RiemannCartan spacetime with zero torsion, there is an exotic brother of the Riemann spacetime, socalled Weitzenböck spacetime, that is characterized by zero total curvature. That means that gravitation in the Weitzenböck spacetime is carried only by torsion, e.g. in the HayashiShirafuji theory [101] and teleparallel gravity [68, 6]. It is even possible to extend the geometry more generally than the RiemannCartan spacetime, as illustrated by Figure 1, and use more spacetime degrees of freedom to gravitate differently.
The third method is to generalize the dimensionality of the spacetime. Spacetime with extra dimensions was first considered by Kaluza in 1919 and Klein in 1926. Despite the failure of their old theories, modern versions of KaluzaKlein theory continue to attract attention. A typical example is the abovementioned DGP theory which exploits the perspective that the ordinary world is a (3+1)D brane to which electromagnetism, the strong and the weak forces are confined, with gravitation extending into the (4+1)D bulk.
Theories in all of the above three categories might explain away dark matter or dark energy, or may be of exotic phenomenological interest. In this thesis, we will focus on the first two categories, in particular gravity and torsion theory, and give further introductions in more details below.
1.3 gravity
There are two important classes of theories: massive theories and dark energy (DE) theories. Interestingly, both classes were motivated by two accelerating eras in the universe. Massive theories, namely polynomials , contain higher order corrections that dominate over the linear GR Lagrangian in the early universe, as the curvature was presumably larger in the past. More subtly, an theory is equivalent to a scalartensor gravity theory, and in the massive case, the emergent scalar field can roll down the emergent potential, which drives inflation at early times. In contrast, the DE branch, exemplified by , is motivated by explaining dark energy that causes the latetime cosmic acceleration. Naively, since is small at late times, negative powers of dominate over the linear GR Lagrangian, and the emergent scalar field can have negative pressure, thus driving the latetime acceleration and explaining dark energy.
However, the archetypal DE model, for [54], where is the Hubble constant at today, suffers from serious problems. First, the theory does not pass solar system tests [60, 73, 76, 81]. Although the Schwarzschild metric can naively solve the field equations for this theory, it can be shown that it is not the solution that satisfies the correct boundary conditions. In fact, it has been shown that the solution that satisfies both the field equations and the correct boundary conditions has the PPN parameter , so this theory is ruled out by, e.g., Shapiro time delay, and deflection of light. Second, the cosmology for this theory is inconsistent with observation when nonrelativistic matter is present [10].
Does this mean that DE theories are dead? The answer is no. In Chapter 3, we exploit the socalled Chameleon effect, which uses nonlinear effects from a very specific singular form of the potential to hide the scalar field from current tests of gravity. In other words, the Chameleon DE models are still consistent with both solar system tests and the latetime cosmic acceleration. We will constrain the gravitational parameters that parametrize the departure from GR in the Chameleon DE models, using solar system tests and cosmological tests in Chapter 3.
1.4 Torsion theories
Theory  Dynamical DOF  Vacuum  Source  Ref.  Notes 

theory 
,  N  Spin  [108]  
Pagels theory  gauge fields  N  Spin  [194]  An gauge theory of gravity 
Metricaffine gravity  General gauge fields  P  Spin  [106]  A gauge theory of gravity in the metricaffine space 
StelleWest  gauge fields  P  Spin, Gradient of the Higgs field  [236]  A gauge theory of gravity spontaneously broken to 
HayashiShirafuji  Tetrads  P  Spin, Mass, Rotation  [101]  A theory in Weitzenböck space 
EinsteinHayashiShirafuji  Tetrads  P  Spin, Mass, Rotation  [154]  A class of theories in RiemannCartan space 
Teleparallel gravity  Tetrads  P  Spin, Mass, Rotation  [68, 6]  A theory in Weitzenböck space 

As illustrated in Figure 1, for the most general manifold with a metric and a connection , departures from Minkowski space are characterized by three geometrical entities: nonmetricity (), curvature () and torsion (), defined as follows:
(1)  
(2)  
(3) 
In GR, spacetime is distorted only by curvature, restricting nonmetricity and torsion to vanish. In RiemannCartan spacetime, gravitation is manifested in the terms of nonzero torsion as well as curvature. There have been many attempts to construct gravitational theories involving torsion, as shown in Table 2. However, testing torsion in the solar system was not a popular idea in the oldfashioned theories for the following two reasons. First, in some torsion theories, e.g. theory [108] and Pagels theory [194], the field equations for the torsion are in algebraic
Whether torsion does or does not satisfy these pessimistic assumptions depends on what the Lagrangian is, which is of course one of the things that should be tested experimentally rather than assumed. Taken at face value, the HayashiShirafuji Lagrangian [101] and teleparallel gravity provide an explicit counterexample to both assumptions, with even a static massive body generating a torsion field. They show that one cannot dismiss out of hand the possibility that mass and angular momentum sources nonlocal torsion (see also Table 2). In Chapter 2, we show that gyroscope experiments such as Gravity Probe B are perfect for testing torsion in nonconventional torsion theories in which torsion can be sourced by rotational angular momentum and can affect the precession of a gyroscope.
After the work in Chapter 2 [154] was published, it has both generated interest [7, 8, 48, 121, 153, 201, 210] and drawn criticism [91, 103, 107]. The controversies are on two levels. On the technical level, in [154] we developed as an illustrative example a family of tetrad theories, the socalled EinsteinHayashiShirafuji (EHS) Lagrangian, in RiemannCartan space which linearly interpolates between GR and the HayashiShirafuji theory. After we submitted the first version of [154], Flanagan and Rosenthal [91] pointed out that the EHS Lagrangian has serious defects. More specifically, in order for the EHS Lagrangian to be a consistent theory (i.e. ghostfree and having wellposed initial value formulation), the parameters of the EHS Lagrangian need to be carefully preselected, and in addition the torsion tensor needs to be minimally coupled to matter. Satisfying these requirements, however, results in a theory that violates the “action equals reaction” principle. Ultimately, then, the EHS Lagrangian does not yield a consistent theory that is capable of predicting a detectable torsion signal for gyroscope experiments. It is worth noting, however, that Flanagan and Rosenthal paper [91] leaves open the possibility that there may be other viable Lagrangians in the same class (where spinning objects generate and feel propagating torsion). The EHS Lagrangian should therefore not be viewed as a viable physical model, but as a pedagogical toy model giving concrete illustrations of the various effects and constraints that we discuss.
On the level of perspectives, Hehl [103] argued that orbital angular momentum density is not a tensor in the field theory since the orbital angular momentum depends on the reference point and the point where momentum acts. Therefore the orbital (and rotational) angular momentum cannot be the source of torsion. In addition, using the multipole exansion method and conservation laws from Noether’s theorem, Puetzfeld and Obukhov [210] argued that nonRiemannian spacetime can only be detected by test particles with intrinsic spins. Their arguments altogether imply that there must be zero torsion in the solar system (no source), and that the GPB gyroscopes, since they have no net polarization, cannot register any signal due to torsion (no coupling). From our point of view, however, the questions of torsion source and coupling have not yet been rigorously settled. The spirit behind our work in [154] is that the answers to these difficult questions can and should be tested experimentally, and that it never hurts to place experimental constraints on an effect even if there are theoretical reasons that favor its nonexistence. The history of science is full of theoretically unexpected discoveries. An example is the discovery of high temperature superconductivity in ceramic compounds containing copperoxide planes: only in metals and metal alloys that had been cooled below 23 K had superconductivity been observed before the mid1980s, but in 1986 Bednorz and Müller [180] discovered that the lanthanum copper oxide, which is an insulator, becomes a superconductor with a high transition temperature of 36 K when it is doped with barium. In the same spirit, we feel that it is valuable to constrain the torsion parameters using the GPB data, despite the nonexistence arguments mentioned above.
2 Cosmology and 21cm tomography
2.1 Cosmological parameters
Thanks to the spectacular technological advancements in circuits and computers, modern cosmologists are fortunate to live in the era of precision cosmology. Using the avalanche of astrophysical data from CMB experiments, large scale galaxy surveys, Type IA supernovae, Ly forest, gravitational lensing and future probes (e.g. 21cm tomography), cosmologists can constrain cosmological parameters to unprecedented accuracies, and in the future may even be able to measure cosmological functions in addition to parameters. In this section, we will give an overview of cosmological parameters, also summarized in Table 3.
Just like there is a concordance theory — GR — in the area of gravitation, there is a concordance model — the standard cosmological model with inflation — in cosmology, successfully parametrized in terms of merely six cosmological parameters. The standard cosmological model is based on the following assumptions:

On large scales, the universe is spatially homogeneous and isotropic (i.e. invariant under translation and rotation) and density fluctuations are small.

The correct gravitational theory is GR.

The universe consists of ordinary baryonic matter, cold nonbaryonic dark matter, dark energy, and electromagnetic and neutrino background radiation.

The primordial density fluctuations are seeded during an inflationary epoch in the early universe.
By the first assumption, the intimidating nonlinear partial differential equations of GR can be accurately solved by using Taylor expansions to linear order in the density fluctuations. Thus, the full description of cosmology consists of two parts: zeroth order (ignoring fluctuations) and linear order (concerning perturbations).
Parameter  Value  Meaning  Definition 
Matter budget parameters:  
Total density/critical density  
Dark energy density parameter  kgm  
Baryon density  kgm  
Cold dark matter density  kgm  
Massive neutrino density  kgm  
Dark energy equation of state  (approximated as constant)  
Seed fluctuation parameters:  
Scalar fluctuation amplitude  Primordial scalar power at /Mpc  
Tensortoscalar ratio  Tensortoscalar power ratio at /Mpc  
Scalar spectral index  Primordial spectral index at /Mpc  
Tensor spectral index  assumed  
Running of spectral index  (approximated as constant)  
Nuisance parameters:  
Reionization optical depth  
Galaxy bias factor  on large scales, where refers to today.  
Other popular parameters (determined by those above):  
Hubble parameter  
Matter density/critical density  
Baryon density/critical density  
CDM density/critical density  
Neutrino density/critical density  
Spatial curvature  
Matter density  
Dark matter neutrino fraction  
Tensor fluctuation amplitude  
eV  Sum of neutrino masses  [135]  
WMAP3 normalization parameter  scaled to /Mpc: if  
Tensortoscalar ratio (WMAP3)  Tensortoscalar power ratio at /Mpc  
Density fluctuation amplitude  , Mpc  
Velocity fluctuation amplitude  
Cosmic history parameters:  
Matterradiation Equality redshift  
Recombination redshift  given by eq. (18) of [115]  
Reionization redshift (abrupt)  (assuming abrupt reionization; [241])  
Acceleration redshift  if  
Myr  Matterradiation Equality time  ( Gyr) [135]  
Myr  Recombination time  ( Gyr) [135]  
Gyr  Reionization time  ( Gyr) [135]  
Gyr  Acceleration time  ( Gyr) [135]  
Gyr  Age of Universe now  ( Gyr) [135]  
Fundamental parameters (independent of observing epoch):  
Primordial fluctuation amplitude  K  
Dimensionless spatial curvature [240]  
Dark energy density  kgm  
Halo formation density  
eV  Matter mass per photon  
eV  Baryon mass per photon  
eV  CDM mass per photon  
eV  Neutrino mass per photon  
Baryon/photon ratio  
Expansion during matter domination  [244]  
Seed amplitude on galaxy scale  Like but on galactic () scale early on 
Zeroth order: the cosmic expansion To zeroth order, the metric for a spatially homogeneous and isotropic universe is completely specified by the socalled FriedmannRobertsonWalker(FRW) line element,
(4) 
which has only one free function , describing the expansion of the universe over time, and one free parameter , the curvature of the 3D space. The Hubble parameter is defined as where the redshift is . The Hubble parameter is both more closely related to observations, and determined by the Friedmann equation
(5) 
obtained by applying the EFE to the FRW metric and a perfect fluid with density and pressure . Here is Newton’s gravitational constant. The Hubble parameter today is usually written where is a unitless number parametrizing our ignorance. The measured value is from WMAP+SDSS data [242].
Cosmological parameters and their measured values are summarized in Table 3. A critical density can be defined such that a universe with total current density equal to is flat (). The matter budget of the universe can be quantified by dimensionless parameters as follows: total matter density , baryonic matter density , dark matter density , massive neutrino density , electromagnetic radiation density , and spatial curvature . The subscript “0” denotes the value at the present epoch. The simplest model for dark energy is a cosmological constant (c.c.) , or the vacuum energy, corresponding to the parameter . A popular approach to generalizing the c.c. is to assume that the equation of state for dark energy is constant.
These parameters are not all independent, e.g. ( is negligible) and . The mathematically equivalent quantities more closely related to observations are , , , , dark matter neutrino fraction , and sum of neutrino masses , since these quantities are simply proportional to the corresponding densities. The energy density of these components have simple dependences on redshift: , , , and . Thus, the Friedmann equation relates the Hubble parameter to these unitless matter budget parameters,
(6) 
First order: the density fluctuations To linear order, perturbations come in two important types: gravitational waves and density fluctuations. The former propagate with the speed of light without growing in amplitude. The latter, however, can get amplified by gravitational instability, and are therefore responsible for structure formation. Density fluctuations are so far observationally consistent with having uncorrelated Gaussiandistributed amplitudes. It is therefore sufficient to use a single function, the socalled power spectrum which gives the variance of the fluctuations as a function of wavenumber and redshift , to characterize the firstorder density perturbations. In principle, can be computed by solving linearized EFE that involves fluctuations in the metric, energy density, pressure, and sometimes shear. In general, depends on three things:

The cosmic matter budget

The seed fluctuations in the early universe

Galaxy formation, including reionization, bias, etc.
In the currently most popular scenario, a large and almost constant energy density stored in a scalar field caused an exponentially rapid expansion at perhaps seconds during a period known as inflation. The theory of inflation can successfully predict negligible spatial curvature (), and solve the horizon problem that the last scattering surface was naively out of causal contact in the noninflationary standard model while the cosmic microwave background radiation (CMBR) is highly spatially homogeneous and isotropic (). Furthermore, inflation can stunningly explain where seed density fluctuations were created: microscopic quantum fluctuations in the aftermath of the Big Bang were stretched to enormous scales during the inflationary epoch. After inflation ended, these seed fluctuations grew into the observed galaxies and galaxy clustering patterns by gravitational instability. The theory of inflation generically predicts almost Gaussiandistributed primordial fluctuations and a nearly scale invariant () adiabatic scalar power spectrum with subdominant gravitational waves. In typical inflation models, the initial power spectrum can be written in the approximate form
(7) 
for the fluctuations in the gravitational potential. Here is the scalar fluctuation amplitude, and the scalar spectral index, at . The minimal set of cosmological parameters approximates to be constant. In a conservative extension, runs linearly in , i.e.
(8) 
where , the logarithmic running of the tilt, is approximated as a constant. In addition to scalar perturbations, the tensor perturbations, related to subdominant gravitational waves, were seeded with the initial power spectrum written in the same form as Eq. (7) except for and replaced by the tensor fluctuation amplitude and the tensor spectral index , respectively. A quantity more closely related by observations is the tensortoscalar ratio .
When seed fluctuations grow into stars, galaxies and galaxy clustering patterns, a number of complicated astrophysical processes are triggered by the structure formation and may influence the clumpiness. For example, during the Epoch of Reionization (), the newlyformed PopIII stars emitted Ly photons, and xrays that reionized neutral hydrogen atoms in the intergalactic medium. Some microwave background photons that have propagated during billions of years from the distant last scattering surface were scattered from the intervening free electrons, generating more anisotropies in the CMBR through the socalled SunyaevZeldovich effect. As a consequence, the CMB power spectrum is sensitive to an integrated quantity known as the reionization optical depth . The Epoch of Reionization is one of the most poorly understood epochs in the cosmic evolution and is therefore of particular interest to cosmologists and astrophysicists.
In addition to reionization, the power spectrum of density fluctuations for galaxies or gas depends on the linear bias . Ordinary baryonic matter cannot gravitate enough to form the observed clumpy structure such as galaxies. In the currently most popular scenario, instead, the observed galaxies trace dark matter halos. As a result, the observed power spectrum from galaxy surveys should be closely related to the real matter power spectrum. A simple widely used model is that on large scales.
CDM model As discussed above, the power spectrum of density fluctuations depends on the cosmic matter budget, the seed fluctuations and nuisance astrophysical parameters. It is striking that the concordance model can fit everything with a fairly small number of cosmological parameters. In this model, the cosmic matter budget consists of about 5% ordinary matter, 30% cold dark matter, hot dark matter (neutrinos) and 65% dark energy. The minimal model space, socalled vanilla set, is parametrized by , setting and . We show a comprehensive set of cosmological parameters in Table 3.
2.2 A brief history of the universe
Cosmic plasma According to the Big Bang theory, the early universe was filled with hot plasma whose contents evolved over time through a series of phase transitions. In the very early universe, the particle constituents were all types of particles in the Standard Model (SM) of particle physics, unidentified dark matter (DM) particles from some extended model of particle physics beyond the SM (e.g. lightest supersymmetric particle and/or axions), and an equal amount of all corresponding antiparticles. The universe cooled as it expanded. When the thermal energy of the cosmic plasma dropped roughly below the rest energy of DM particles, DM particles froze out (at for typical WIMPs) and have not been in thermal equilibrium with other constituents since. DM particles eventually became an almost collisionless and cold (nonrelativistic) component that constitutes about 20% of the cosmic matter budget at the present day.
As the cosmic temperature kept decreasing, the symmetry between baryons and antibaryons was broken at . The tiny asymmetry at the level of was followed by matterantimatter annihilation, forming protons that constitute about 4% of the cosmic matter budget at the present day. This is a hypothetical process known as baryogenesis. After the baryogenesis, the cosmic hot soup was a cauldron of protons, electrons and photons, and a smattering of other particles (e.g. hot neutrinos).
When the universe was cooled to below about 1 MeV – the mass difference between a neutron and a proton – neutrons froze out at as weak interactions like ceased. Subsequently, protons and neutrons combined to form light element such as deuterium ( or ), tritium () and helium ( and ) in a process known as big bang nucleosynthesis (BBN). For example, deuterium forms via ; then , after which . The helium nucleus () is the most stable among light elements, and after BBN, about 75% of baryons in the universe are hydrogen nuclei (i.e. protons), while nearly 25% are helium nuclei.
The freelymoving electrons tightly coupled to photons via Compton scattering and electrons to protons and other nuclei via Coulomb scattering, keeping the cosmic plasma in equilibrium. All components except for photons were in the form of ions until temperatures fell to Kelvin, when protons and electrons combined to form electrically neutral hydrogen atoms — a process known as recombination. The photons at that temperature were no longer energetic enough to reionize significant amounts of electrons. The Compton scattering process therefore ended, decoupling the photons from the matter. Thus, the cosmos become almost transparent to photons, releasing the microwave background. The gas temperature continues to drop as the universe expands, so one might expect that the cosmic gas would still be cold and neutral today.
Surprisingly, it is not. To understand why, we take a detour and first review how galaxies form, and come back to this question subsequently.
Galaxy formation According to the current most popular scenario, at , the universe underwent a period of inflation. The cosmic inflation stretched the universe by 55 efoldings, i.e. a lattice grid was more than times larger than itself before inflation, making the universe extremely flat. After inflation, the universe was approximately spatially homogeneous and isotropic because particles at any two largely separated points that otherwise could by no means have causal contact without inflation may actually be in close together during the inflation, and equilibrate their temperatures through the exchange of force carriers that would have had time to propagate back and forth between them.
Cosmic inflation also created seed fluctuations at the level of one part in a hundred thousand in the early universe. After the end of inflation, however, the universe was dominated by radiation, i.e. ultrarelativistic particles that moved fast enough to keep these primordial density fluctuations from growing. Fortunately, the energy density of radiation dropped more rapidly than matter density as the universe expanded; quantitatively, for radiation and for matter. Consequently, at years, matter became the dominant component of the universe, and the fluctuations began to grow due to gravitational instability — which means that a region that started slightly denser than average pulled itself together by its own gravity. More specifically, the denser region initially expanded with the whole universe, but its extra gravity slowed its expansion down, turned it around and eventually made the region collapse on itself to form a bound object such as a galaxy.
Reionization: cosmic plasma revisited Now we come back to the question: is the present universe filled with mostly neutral hydrogen atoms? Although the terrestrial world is composed of atoms, the intergalactic medium hosts the majority of ordinary matter in the form of plasma. Conclusive evidence comes from two types of observations. The Wilkinson Microwave Anisotropy Probe (WMAP) and other experiments have confirmed that the CMBR is slightly polarized (in socalled EE modes). Since only free electrons (and not neutral hydrogen atoms) scatter and polarize this radiation, the amount of polarization observed on large angular scales suggests that the neutral gas was reionized into plasma as early as a few hundred million years after our big bang. Independent confirmation of cosmic reionization come from the observed spectra of the distant quasars that indicates that reionization should be complete by a billion years after the big bang.
The details of cosmic reionization are still a blank page that needs to be filled by upcoming observations. There are, however, some plausible pictures that reside in the minds of theorists. In the current models, the oldest galaxies are dwarf galaxies that started to form at a cosmic age of a few hundred million years. Larger galaxies such as the Milky Way were latecomers that were born from the gradual coalescence of many dwarf galaxies. Stars were created when the gas in embryonic galaxies got cool and fragmented. The first generation of stars, socalled PopIII stars, triggered the nuclear fusion of hydrogen and released energy in the form of ultraviolet photons in amounts a million times larger than the energy needed to ionize the same mass of neutral gas ( for each hydrogen atom). The emitted ultraviolet photons leaked into the intergalactic medium, broke the neutral hydrogen atoms back down into their constituent protons and electrons, and created an expanding bubble of ionized gas. As new galaxies took root, more bubbles appeared, overlapped and eventually filled all of intergalactic space.
Some researchers conjecture that black holes rather than stars may have caused cosmic reionization. Like stars, black holes arise from galaxies. Particles that plummeted into black holes emitted xrays in an amount of energy 10 million times larger than the ionization energy of the same amount of hydrogen. The mechanisms of reionization by massive stars or black holes can be distinguished by observing the boundaries of ionized bubbles in upcoming experiments. Ultraviolet photons emitted by massive stars were easily absorbed by the neutral gas, while xrays from black holes can penetrate deeply into the intergalactic medium, so, black holes are associated with fuzzier bubble boundaries.
Dark Ages Both reionization models predict that the cosmic reionization started to take shape after the first galaxies formed at . Between the release of the microwave background at and the formation of first galaxies, however, there is a tremendous gap! During these socalled Dark Ages (DA), the universe was dark since ordinary matter was in the form of neutral atoms that were not hot enough to radiate light. Since the cosmic matter was transparent to the microwave background photons, the CMB photons no longer traced the distribution of matter. However, the DA were not a boring period. In fact, the DA are an interesting embryonic interlude between the seeding of density fluctuations and the birth of first galaxies: within the inky blackness, the primordial matter clumps grew by their extra gravity and eventually collapsed on themselves into galaxies. The secret of galaxy formation is hidden in the DA.
But how can we probe a period that was by its very nature practically dark? Fortunately, even cold hydrogen atoms emit feeble light with a wavelength of 21 centimeters. Below we describe how observations of the 21cm line are emerging as a promising probe of the epoch of reionization (EoR) and the Dark Ages.
2.3 21cm line: spin temperature
In quantum mechanics, particles carry an intrinsic angular momentum known as spin. For example, a particle with spin such as a proton or electron can have its angular momentum vector point either “up” or “down”. In a hydrogen atom, the interaction between the spins of the nucleus (the proton) and the electron splits the ground state into two hyperfine states, i.e., the triplet states of parallel spins and the singlet state of antiparallel spins. The antiparallel spin state has lower energy than the parallel spin state, and the transition between them corresponds to the emission or absorption of a photon with the wavelength of 21 centimeters. For the 21cm transition, the socalled spin temperature quantifies the fraction of atoms in each of the two states: the ratio of number densities is
(9) 
Here the subscripts 1 and 0 denote the parallel and the antiparallel spin state, respectively. is the number density of atoms in the th state, and is the statistical weight (=3 and =1), is the energy splitting, and is the equivalent temperature.
21cm observations aim to compare lines of sight through intergalactic hydrogen gas to hypothetical sightlines without gas and with clear views of CMB. Thus, one should observe emission lines if , or absorption lines if . Here is the CMB temperature at redshift .
There are three competing mechanisms that drive : (1) absorption of CMB photons; (2) collisions with other hydrogen atoms, free electrons and protons; and (3) scattering of ultraviolet photons. For the first mechanism, the process of absorption of microwave background photons tends to equilibrate the spin temperature with the CMB temperature. For the second, spin exchange due to collisions is efficient when gas density is large. The third mechanism, also known as the WouthuysenField mechanism, involves transitions from one hyperfine state to the first excited state and then down to the other hyperfine state with different spin orientation, which couples the 21cm excitation to the ultraviolet radiation field.
The global history of the intergalactic gas is defined by three temperatures: the spin temperature (a measure of the spin excitation) as defined above, the kinetic temperature of the intergalactic gas (a measure of atomic motions), and the microwave background temperature (a measure of the energy of background photons). These temperatures can approach or deviate from one another, depending on which physical processes are dominant. In a threeway relation (see Figure 3), after an initial period when three temperatures are all equal, spin temperature first traces the kinetic temperature, then the background temperature, and eventually the kinetic temperature again.
Initially after the CMB is released, although the neutral atoms are transparent to the background photons, free electrons left over from recombination mediate the exchange of energy between background photons and atoms via Compton scattering (between photons and free electrons) and Coulomb scattering (between free electrons and hydrogen nuclei). The kinetic temperature therefore tracks the CMB temperature, and also the spin temperature due to collisions between hydrogen atoms. Observations of this period will therefore find neither emission or absorption of 21cm lines against the microwave background.
The first transition took place when the universe was about 10 million years old. As the universe expanded, the gas was diluted and cooled, and the free electron mediation eventually became so inefficient that the atomic motions decoupled from the background radiation at a redshift of about 150 and underwent adiabatic cooling at a more rapid rate () than the cooling of the CMB (). In this phase, the spin temperature matched the kinetic temperature due to collisions, and neutral gas absorbed the background photons.
When the universe was close to a hundred million years old, a second transition occurred. As the gas continued to expand, collisions between atoms became infrequent, and made the coupling between kinetic temperature and spin temperature inefficient. As a consequence, the spin excitation reached equilibrium with the background photons again. Thus, we cannot observe gas from this period.
After the first stars and quasars lit up, a third transition occurred. The intergalactic gas was heated up by ultraviolet photons, xrays or shocks from galaxies. In addition, spin exchange through the scattering of ultraviolet photons became important, coupling spin temperature back to approximately the kinetic temperature. Since flipping the spins takes much less energy than ionizing atoms, neutral gas began to glow in 21cm radiation well before becoming ionized. Finally, as the hydrogen became fully reionized, the 21cm emission faded away.
2.4 21cm cosmology
The threeway relation between , and determines whether absorption or emission lines, or neither, of 21cm signals can be detected against the microwave background. However, the observed quantities from 21cm experiments are not these temperatures. In this section we will describe how to extract cosmological information from the 21cm signal.
The observable in 21cm experiments is the difference between the observed 21 cm brightness temperature at the redshifted frequency and the CMB temperature , given by [88]
(10) 
where is number density of the neutral hydrogen gas, and s is the spontaneous decay rate of 21cm excitation. The factor is the gradient of the physical velocity along the line of sight ( is the comoving distance), which is on average (i.e. for no peculiar velocity). Here is the Hubble parameter at redshift .
21cm experiments can measure the statistical properties (such as power spectrum) of brightness temperature and even map . The brightness temperature is determined by four quantities — hydrogen mass density, spin temperature, neutral fraction, and peculiar velocity. Among them, only fluctuations in the hydrogen mass density can be used to test cosmological models, and how to disentangle density fluctuations from other quantities remains an open question. Although fluctuations in are poorly understood, this complication can be circumvented using the fact that the factor is saturated to be unity when . This condition is usually satisfied, since the gas should be heated enough by ultraviolet photons, xrays and shocks before and during the reionization. Consequently, the fluctuations in the factor can be neglected.
Fluctuations in neutral fraction are important during reionization, and are unfortunately also poorly understood. In order to effectively use 21cm lines as probes of cosmology, two solutions to the problem of how to disentangle matter density fluctuations from the fluctuations in neutral fraction have been proposed in the past.

Since flipping the spin takes much less energy than ionizing an atom, it is plausible that there exists a prereionization period in which both and hold. In this period, the matter power spectrum dominates the total power spectrum.

As long as density fluctuations are much smaller than unity on large scales, linear perturbation theory will be valid, so the peculiar velocity can be used to decompose the total 21cm power spectrum into parts with different dependencies on caused by the socalled redshift space distortion, where is the cosine of the angle between the Fourier vector and the line of sight. Only the forth moment in the total power spectrum, i.e. a term containing , depends on the matter power spectrum alone, and all other moments are contaminated by power spectra related to fluctuations in neutral fraction. One can in principle separate the term from the contaminated terms, and use only it to constrain cosmology.
In Chapter 4, we will develop a third method that exploits the smoothness of the nuisance power spectra and parametrizes them in terms of seven constants at each redshift. Thus, the combination of cosmological parameters and nuisance parameters completely dictate total power spectrum. This socalled MID method turns out to be as effective as the simplest methods for longterm 21cm experiments, but more accurate.
2.5 Prospects of 21cm tomography
By observing 21cm signal from a broad range of epochs, neutral intergalactic hydrogen gas can be mapped by upcoming 21cm experiments. This approach, known as 21cm tomography, is emerging as one of the most promising cosmological probes for the next decades, since it encodes a wealth of information about cosmology, arguably even more than the microwave background. The reasons behind this optimistic perspective are as follows.
First, mapping of neutral hydrogen can be done over a broad range of frequencies corresponding to different redshifts, and is therefore threedimensional, with the third dimension along the line of sight. In contrast, the twodimensional microwave background is a map of anisotropy of radiations in the sky from the last scattering surface, a narrow spherical shell at the epoch of recombination. The 3D mapping, in principle, measures a much larger number of modes than 2D mapping, and therefore has the potential to measure the matter power spectrum and cosmological parameters with less sample variance.
Second, the range of 21cm tomography goes from the dark ages to the epoch of reionization, which is almost a complete time line of galaxy formation. Mapping of neutral hydrogen along this time line provides an observational view of how primordial density fluctuations evolved to form galaxies, a picture that has hitherto only existed in theorists’ minds.
Third, 21cm tomography contains information not only about the matter density fluctuations that seeded galaxies, but also on the effects that the galaxies, after their formation, had on their surroundings, e.g. reionization and heating of the intergalactic gas, etc. Separating physics (matter power spectrum) from astrophysics (ionization power spectrum, power spectrum of spin temperature fluctuations) can be used not only to constrain cosmology, but to learn about astrophysical process.
Last but not the least, 21cm tomography can shed light on testing fundamental particle physics and gravitational physics. During the dark ages, the spin temperature traces the kinetic temperature by collisions of neutral atoms or microwave background temperature by absorption of CMB photons. Since no complicated astrophysics (e.g. reionization) takes effect during the dark ages, the dark ages are a well controlled cosmic laboratory. Nonstandard particle physics models may have unusual decay of dark matter which imprints a signature on the dark ages. Also, many modified gravitational theories can be distinguished by their predictions for galaxy formation.
However, observers will have to overcome a great deal of challenges. Firstly, the redshifted 21cm signals fall in the lowfrequency radio band, from to . Thus, lowfrequency radio broadcasts on Earth must be filtered out. In fact, most 21cm experiments (except LOFAR) have chosen their sites at lowpopulation spots. Secondly, thermal noise is approximately proportional to the wavelength to roughly the 2.6 power, because of synchrotron radio from our own galaxy. Noise at the ultra low frequency side will therefore overwhelm the signal from the dark ages, making observation of the dark ages technically unrealistic with the upcoming first generation of 21cm experiments. Even at the higher frequencies corresponding to the epoch of reionization, synchrotron foreground is about four orders of magnitude more intense than the cosmic signal. Fortunately, the foreground spectra are smooth functions of wavelength and may vary slowly, allowing them to be accurately subtracted out.
To detect the 21cm signal, four first generation observatories — the Murchison Widefield Array (MWA) [182], the 21 Centimeter Array (21CMA) [1], the Low Frequency Array (LOFAR) [148] and the Precision Array to Probe Epoch of Reionization (PAPER) [196] — are currently under development. The next generation observatory, Square Kilometre Array (SKA) [226], is in the fundraising and design stage. Furthermore, 21cm tomography optimized square kilometer array known as the Fast Fourier Transform Telescope (FFTT) [245], which has been forecast to be capable of extremely accurate cosmological parameters measurements, has been proposed. The next two decades should be a golden age for 21cm tomography, both observationally and theoretically.
3 Road map
The rest of this thesis is organized as follows. In Chapter 2, we parametrize the torsion field around Earth, derive the precession rate of GPB gyroscopes in terms of the abovementioned modelindependent parameters, and constrain the torsion parameters with the ongoing GPB experiment together with other solar system tests. We also present the EHS theory as a toy model of an angularmomentum coupled torsion theory, and constrain the EHS parameters with the same set of experiments. The work in Chapter 2 has been published in Physical Review D [154]. In Chapter 3, after a review of the equivalence of theories with scalar tensor theories, we explore the Chameleon model and massive theories, respectively, focusing on observational constraints. The work in Chapter 3 has been published in Physical Review D [86]. In Chapter 4, we explain the assumptions that affect the forecast of cosmological parameter measurements with 21cm tomography, and also present a new method for modeling the ionization power spectra. We quantify how the cosmological parameter measurement accuracy depends on each assumption, derive simple analytic approximations of these relations, and discuss the relative importance of these assumptions and implications for experimental design. The work in Chapter 4 has been accepted for publication in Physical Review D [155]. In Chapter 5, we conclude and discuss possible extensions to the work in the thesis.
The contributions to the work in this thesis are as follows. For [154], I carried out all detailed calculations and plots. Max Tegmark initially suggested the idea of constraining torsion with GPB, and he and Alan Guth were extensively involved in the discussion of results. Serkan Cabi contributed to the discussion of generalized gravitational theories. For [86], Tom Faulkner carried out all detailed calculations and plots. I checked and corrected preliminary results. Max Tegmark initially suggested the idea of constraining viable theories, and he and Ted Bunn were extensively involved in the discussion of results. For [155], I did the bulk of the work, including writing analysis software, inventing the MID model parametrization of nuisance power spectra, performing the calculations and consistency checks. Max Tegmark initially suggested the idea of investigating how forecasts of 21cm tomography depend on various assumptions and was extensively involved in discussions of the results as the project progressed. Matt McQuinn contributed his radiative transfer simulation results, and he, Matias Zaldarriaga and Oliver Zahn also participated in detailed discussions of results and strategy. Oliver Zahn also helped with consistency checks of the Fisher matrix results.
Chapter \thechapter Constraining torsion with Gravity Probe B
4 Introduction
Einstein’s General Theory of Relativity (GR) has emerged as the hands down most popular candidate for a relativistic theory of gravitation, owing both to its elegant structure and to its impressive agreement with a host of experimental tests since it was first proposed about ninety years ago [267, 265, 268]. Yet it remains worthwhile to subject GR to further tests whenever possible, since these can either build further confidence in the theory or uncover new physics. Early efforts in this regard focused on weakfield solar system tests, and efforts to test GR have since been extended to probe stronger gravitational fields involved in binary compact objects, black hole accretion and cosmology [118, 259, 260, 58, 204, 232, 233, 112, 113, 250, 15, 57, 163, 162, 199, 229, 230, 228, 5, 134, 9, 119, 161, 82, 83, 67, 209, 124, 35, 179, 26, 151].
4.1 Generalizing general relativity
The arguably most beautiful aspect of GR is that it geometrizes gravitation, with Minkowski spacetime being deformed by the matter (and energy) inside it. As illustrated in Figure 1, for the most general manifold with a metric and a connection , departures from Minkowski space are characterized by three geometrical entities: nonmetricity (), curvature () and torsion (), defined as follows:
(11)  
(12)  
(13) 
GR is the special case where the nonmetricity and torsion are assumed to vanish identically (, i.e., Riemann spacetime), which determines the connection in terms of the metric and leaves the metric as the only dynamical entity. However, as Figure 1 illustrates, this is by no means the only possibility, and many alternative geometric gravity theories have been discussed in the literature [36, 99, 96, 102, 68, 6, 108, 186, 258, 49, 94, 223, 230, 17, 207, 167, 131, 253, 170, 190, 248, 128, 97, 105, 206, 106, 149, 104, 211, 256, 60, 98, 31, 141, 214, 11, 47, 85, 53] corresponding to alternative deforming geometries where other subsets of vanish. Embedding GR in a broader parametrized class of theories allowing nonvanishing torsion and nonmetricity, and experimentally constraining these parameters would provide a natural generalization of the highly successful parametrized postNewtonian (PPN) program for GR testing, which assumes vanishing torsion [267, 265, 268].
For the purposes of this chapter, a particularly interesting generalization of Riemann spacetime is RiemannCartan Spacetime (also known as ), which retains but is characterized by nonvanishing torsion. In , torsion can be dynamical and consequently play a role in gravitation alongside the metric. Note that gravitation theories including torsion retain what are often regarded as the most beautiful aspects of General Relativity, i.e. general covariance and the idea that “gravity is geometry”. Torsion is just as geometrical an entity as curvature, and torsion theories can be consistent with the Weak Equivalence Principle (WEP).
4.2 Why torsion testing is timely
Experimental searches for torsion have so far been rather limited [99], in part because most published torsion theories predict a negligible amount of torsion in the solar system. First of all, many torsion Lagrangians imply that torsion is related to its source via an algebraic equation rather than via a differential equation, so that (as opposed to curvature), torsion must vanish in vacuum. Second, even within the subset of torsion theories where torsion propagates and can exist in vacuum, it is usually assumed that it couples only to intrinsic spin, not to rotational angular momentum [108, 237, 272], and is therefore negligibly small far from extreme objects such as neutron stars. This second assumption also implies that even if torsion were present in the solar system, it would only affect particles with intrinsic spin (e.g. a gyroscope with net magnetic polarization) [237, 272, 187, 110, 111, 65, 137, 12], while having no influence on the precession of a gyroscope without nuclear spin [237, 272, 187] such as a gyroscope in Gravity Probe B.
Whether torsion does or does not satisfy these pessimistic assumptions depends on what the Lagrangian is, which is of course one of the things that should be tested experimentally rather than assumed. Taken at face value, the HayashiShirafuji Lagrangian [101] provides an explicit counterexample to both assumptions, with even a static massive body generating a torsion field — indeed, such a strong one that the gravitational forces are due entirely to torsion, not to curvature. As another illustrative example, we will develop in Section 12 a family of tetrad theories in RiemannCartan space which linearly interpolate between GR and the HayashiShirafuji theory. Although these particular Lagrangeans come with important caveats to which we return below (see also [91]), they show that one cannot dismiss out of hand the possibility that angular momentum sources nonlocal torsion (see also Table 2). Note that the proof[237, 272, 187] of the oftrepeated assertion that a gyroscope without nuclear spin cannot feel torsion crucially relies on the assumption that orbital angular momentum cannot be the source of torsion. This proof is therefore not generally applicable in the context of nonstandard torsion theories.
More generally, in the spirit of actionreaction, if a (nonrotating or rotating) mass like a planet can generate torsion, then a gyroscope without nuclear spin could be expected feel torsion, so the question of whether a nonstandard gravitational Lagrangian causes torsion in the solar system is one which can and should be addressed experimentally.
This experimental question is timely because the Stanfordled
gyroscope satellite experiment, Gravity Probe B
4.3 How this chapter is organized
In general, torsion has 24 independent components, each being a function of time and position. Fortunately, symmetry arguments and a perturbative expansion will allow us to greatly simplify the possible form of any torsion field of Earth, a nearly spherical slowly rotating massive object. We will show that the most general possibility can be elegantly parametrized by merely seven numerical constants to be constrained experimentally. We then derive the effect of torsion on the precession rate of a gyroscope in Earth orbit and work out how the anomalous precession that GPB would register depends on these seven parameters.
The rest of this chapter is organized as follows. In Section 5, we review the basics of RiemannCartan spacetime. In Section 6, we derive the results of parametrizing the torsion field around Earth. In Section 7, we discuss the equation of motion for the precession of a gyroscope and the worldline of its center of mass. We use the results to calculate the instantaneous precession rate in Section 8, and then analyze the Fourier moments for the particular orbit of GPB in Section 9. In Section 10, we show that GPB can constrain two linear combinations of the seven torsion parameters, given the constraints on the PPN parameters and from other solar system tests. To make our discussion less abstract, we study HayashiShirafuji torsion gravity as an explicit illustrative example of an alternative gravitational theory that can be tested within our framework. In Section 11, we review the basics of Weitzenböck spacetime and HayashiShirafuji theory, and then give the torsionequivalent of the linearized Kerr solution. In Section 12, we generalize the HayashiShirafuji theory to a twoparameter family of gravity theories, which we will term EinsteinHayashiShirafuji (EHS) theories, interpolating between torsionfree GR and the HayashiShirafuji maximal torsion theory. In Section 13, we apply the precession rate results to the EHS theories and discuss the observational constraints that GPB, alongside other solar system tests, will be able to place on the parameter space of the family of EHS theories. We conclude in Section 27. Technical details of torsion parametrization (i.e. Section 6) are given in Appendices 15 & 16. Derivation of solar system tests are given in Appendix 17. We also demonstrate in Appendix 18 that current groundbased experimental upper bounds on the photon mass do not place more stringent constraints on the torsion parameters or than GPB will.
After the first version of the paper [154] involving the work in this chapter was submitted, Flanagan and Rosenthal showed that the EinsteinHayashiShirafuji Lagrangian has serious defects [91], while leaving open the possibility that there may be other viable Lagrangians in the same class (where spinning objects generate and feel propagating torsion). The EHS Lagrangian should therefore not be viewed as a viable physical model, but as a pedagogical toy model giving concrete illustrations of the various effects and constraints that we discuss.
Throughout this chapter, we use natural gravitational units where . Unless we explicitly state otherwise, a Greek letter denotes an index running from 0 to 3 and a Latin letter an index from 1 to 3. We use the metric signature convention .
5 RiemannCartan spacetime
We review the basics of RiemannCartan spacetime only briefly here, and refer the interested reader to Hehl et al. [108] for a more comprehensive discussion of spacetime with torsion. RiemannCartan spacetime is a connected fourdimensional manifold endowed with metric of Lorentzian signature and an affine connection such that the nonmetricity defined by Eq. (11) with respect to the full connection identically vanishes. In other words, the connection in RiemannCartan spacetime may have torsion, but it must still be compatible with the metric (). The covariant derivative of a vector is given by
(14)  