\thechapter Introduction

Constraining Gravitational and Cosmological Parameters with Astrophysical Data

\department

Department of Physics \degreeDoctor of Philosophy \degreemonthJune \degreeyear2008 \thesisdateMay 23, 2008

\supervisor

Max TegmarkAssociate Professor of Physics \supervisorAlan H. GuthVictor F. Weisskopf Professor of Physics

\chairman

Thomas J. GreytakAssociate Department Head for Education

{abstractpage}

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 scalar-tensor 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 so-called Chameleon Effect. We explore observational constraints from the late-time 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 3-dimensional 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 particle-theory 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:

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 hands-down 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. So-called -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 late-time 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 3-dimensional brane is embedded in a 5-dimensional 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 tensor-vector-scalar (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 so-called Parametrized Post-Friedmann 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 “non-linearity” in the 0 (i) Perihelion shift from helioseismology
superposition law for gravity (ii) Nordtvedt effect
Preferred-location effects 0 Earth tides Gravimeter data
Preferred-frame 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
Table 1: The PPN parameters, their significance and experimental bounds. Contents of this table except for the GPB entry are taken from Table 2 of [268] and Table 1 of [267].

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, space-time 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 non-gravitational 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 well-known weak-field solar system tests: Mercury’s perihelion shift, light bending, gravitational redshift, Shapiro time delay and lunar laser-ranging. An on-going satellite experiment, Gravity Probe B, that measures the spin precession of free-falling 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 (frame-dragging effect) of Earth. Additionally, the LAGEOS [62] experiment has directly confirmed the frame-dragging effect on the orbits of test particles around the rotating Earth.

In the weak-field regime, there exists a mature formalism, the so-called Parametrized Post-Newtonian (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 strong-field binary compact objects and black-hole 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 model-independent 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 weak-field limit with 10 PPN parameters (see Table 1) that can be constrained by solar-system test data. The developing Parametrized Post-Friedmann (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 so-called Riemann spacetime, that is completely specified by the metric , a tensor at each spacetime position. In the Riemann spacetime, a free-falling 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 so-called 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 inverse-square 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 so-called gravity. A new scalar field with arbitrary potential and couplings to the metric can also be introduced into the action — this is so-called scalar-tensor gravity. In fact, gravity is equivalent to a special class of scalar-tensor gravity theories. Additionally, the action can include even more fields (vector fields plus scalar fields), as in the so-called TeVeS (standing for “tensor-vector-scalar”) 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 so-called Riemann-Cartan 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 non-zero torsion relaxes this constraint. The geometry of Riemann-Cartan spacetime is pinned down by the metric and torsion – the so-called torsion theory is established in terms of these two pieces. Just as Riemann spacetime is a special case of Riemann-Cartan spacetime with zero torsion, there is an exotic brother of the Riemann spacetime, so-called 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 Hayashi-Shirafuji theory [101] and teleparallel gravity [68, 6]. It is even possible to extend the geometry more generally than the Riemann-Cartan 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 Kaluza-Klein theory continue to attract attention. A typical example is the above-mentioned 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 scalar-tensor 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 late-time 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 late-time 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 non-relativistic matter is present [10].

Does this mean that -DE theories are dead? The answer is no. In Chapter 3, we exploit the so-called Chameleon effect, which uses non-linear 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 late-time 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

Figure 1: Classification of spaces (Q,R,S) and the reduction flow. Metric-Affine spacetime is a manifold endowed with Lorentzian metric and linear affine connection without any restrictions. All spaces below it except the Weyl-Cartan space are special cases obtained from it by imposing three types of constraints: vanishing non-metricity tensor ( for short), vanishing Riemann curvature tensor ( for short), or vanishing torsion tensor ( for short). A plus sign in a parenthesis indicates a non-vanishing quantity from the set , and a minus sign a vanishing quantity. For example, Riemann spacetime means that but . Weyl-Cartan space is a Metric-Affine space with vanishing “tracefree nonmetricity” ( for short), defined by . The trace of the nonmetricity is defined by ; thus is automatically trace free, i.e., . Subsets of this classification scheme are shown in Fig. 2 of [106], Fig. 1 of [209] and Fig. 5 of [96]. Among the terms, Einstein-Weyl, Weitzenböck and Minkowski spaces are standard, Metric-Affine, Weyl-Cartan, Riemann-Cartan and Riemann spaces follow [106], and we here introduce the terms Weitzenböck-Weyl and Minkowski-Weyl space by symmetry.
Theory Dynamical DOF Vacuum Source Ref. Notes

theory
, N Spin [108]
Pagels theory gauge fields N Spin [194] An gauge theory of gravity
Metric-affine gravity General gauge fields P Spin [106] A gauge theory of gravity in the metric-affine space
Stelle-West gauge fields P Spin, Gradient of the Higgs field [236] A gauge theory of gravity spontaneously broken to
Hayashi-Shirafuji Tetrads P Spin, Mass, Rotation [101] A theory in Weitzenböck space
Einstein-Hayashi-Shirafuji Tetrads P Spin, Mass, Rotation [154] A class of theories in Riemann-Cartan space
Teleparallel gravity Tetrads P Spin, Mass, Rotation [68, 6] A theory in Weitzenböck space

Table 2: A short list of torsion theories of gravity. The “DOF” in the second column is short for “degrees of freedom”. In the column Vacuum, “N” refers to non-propagating torsion in the vacuum while “P” means propagating torsion. In the column Source, “spin” refers to intrinsic spin, “mass” means non-rotating mass, and “rotation” means rotational angular momentum.

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: non-metricity (), curvature () and torsion (), defined as follows:

(1)
(2)
(3)

In GR, spacetime is distorted only by curvature, restricting non-metricity and torsion to vanish. In Riemann-Cartan 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 old-fashioned 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 1 rather than differential form, which means that torsion cannot propagate into the vacuum. Second, it is well-entrenched folklore that the source of the torsion can only be the intrinsic spin of elementary particles, not the mass or the rotational angular momentum of a massive object. That means that, even if torsion could propagate and exist in the vacuum, since the net spin polarizations of the Sun and Earth are negligible, the torsion should still vanish in the solar system. 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) [12, 65, 110, 111, 137, 187, 237, 272], while having no influence on the precession of a gyroscope without nuclear spin [187, 237, 272] such as a gyroscope in Gravity Probe B. The upshot is that any torsion theory that makes either of two above assumptions has already found itself in a position that can never be ruled out by solar system tests. Such a torsion theory can have noticeable effects only in extreme situations, e.g. near neutron stars with net magnetic polarizations.

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 Hayashi-Shirafuji 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 non-local torsion (see also Table 2). In Chapter 2, we show that gyroscope experiments such as Gravity Probe B are perfect for testing torsion in non-conventional 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 so-called Einstein-Hayashi-Shirafuji (EHS) Lagrangian, in Riemann-Cartan space which linearly interpolates between GR and the Hayashi-Shirafuji 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. ghost-free and having well-posed initial value formulation), the parameters of the EHS Lagrangian need to be carefully pre-selected, 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 non-Riemannian 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 non-existence. The history of science is full of theoretically unexpected discoveries. An example is the discovery of high temperature superconductivity in ceramic compounds containing copper-oxide planes: only in metals and metal alloys that had been cooled below 23 K had superconductivity been observed before the mid-1980s, 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 non-existence 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:

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

  2. The correct gravitational theory is GR.

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

  4. The primordial density fluctuations are seeded during an inflationary epoch in the early universe.

By the first assumption, the intimidating non-linear 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
Tensor-to-scalar ratio Tensor-to-scalar 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
Tensor-to-scalar ratio (WMAP3) Tensor-to-scalar power ratio at /Mpc
Density fluctuation amplitude , Mpc
Velocity fluctuation amplitude
Cosmic history parameters:
Matter-radiation Equality redshift
Recombination redshift given by eq. (18) of [115]
Reionization redshift (abrupt) (assuming abrupt reionization; [241])
Acceleration redshift if
Myr Matter-radiation 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
Table 3: Cosmological parameters measured from WMAP and SDSS LRG data. Error bars are . This table is adapted from Table 2 of [242].

Zeroth order: the cosmic expansion To zeroth order, the metric for a spatially homogeneous and isotropic universe is completely specified by the so-called Friedmann-Robertson-Walker(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 Gaussian-distributed amplitudes. It is therefore sufficient to use a single function, the so-called power spectrum which gives the variance of the fluctuations as a function of wavenumber and redshift , to characterize the first-order 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:

  1. The cosmic matter budget

  2. The seed fluctuations in the early universe

  3. 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 non-inflationary 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 Gaussian-distributed 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 tensor-to-scalar 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 newly-formed Pop-III stars emitted Ly photons, and x-rays that re-ionized neutral hydrogen atoms in the inter-galactic 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 so-called Sunyaev-Zeldovich 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, so-called 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 anti-particles. 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 (non-relativistic) 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 anti-baryons was broken at . The tiny asymmetry at the level of was followed by matter-antimatter 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 freely-moving 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 re-ionize 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.

Figure 2: Cosmic time line: a brief history of the universe from the aftermath of the big bang to the present day.

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 e-foldings, 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. ultra-relativistic 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 so-called 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, so-called Pop-III 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 x-rays 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 x-rays 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 so-called 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 anti-parallel spins. The anti-parallel 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 so-called 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 anti-parallel 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 Wouthuysen-Field 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 three-way 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.

Figure 3: Thermal history of the intergalactic gas

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, x-rays 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 three-way 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, x-rays 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.

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

  2. 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 so-called 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 so-called MID method turns out to be as effective as the simplest methods for long-term 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 three-dimensional, with the third dimension along the line of sight. In contrast, the two-dimensional 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. Non-standard 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 low-frequency radio band, from to . Thus, low-frequency radio broadcasts on Earth must be filtered out. In fact, most 21cm experiments (except LOFAR) have chosen their sites at low-population 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 fund-raising 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 above-mentioned model-independent 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 angular-momentum 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 weak-field 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: non-metricity (), curvature () and torsion (), defined as follows:

(11)
(12)
(13)

GR is the special case where the non-metricity 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 non-vanishing torsion and non-metricity, and experimentally constraining these parameters would provide a natural generalization of the highly successful parametrized post-Newtonian (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 Riemann-Cartan Spacetime (also known as ), which retains but is characterized by non-vanishing 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 Hayashi-Shirafuji 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 Riemann-Cartan space which linearly interpolate between GR and the Hayashi-Shirafuji 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 non-local torsion (see also Table 2). Note that the proof[237, 272, 187] of the oft-repeated 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 non-standard torsion theories.

More generally, in the spirit of actionreaction, if a (non-rotating 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 non-standard gravitational Lagrangian causes torsion in the solar system is one which can and should be addressed experimentally.

This experimental question is timely because the Stanford-led gyroscope satellite experiment, Gravity Probe B2 (GPB), was launched in April 2004 and has successfully been taking data. Preliminary GPB results, released in April 2007, have confirmed the geodetic precession to better than 1%, and the full results, which are highly relevant to this chapter, are due to be released soon. GPB contains a set of four extremely spherical gyroscopes and flies in a circular polar orbit with altitude 640 kilometers, and we will show that it has the potential to severely constrain a broad class of previously allowed torsion theories. GPB was intended to test the GR prediction [219, 266, 4, 34, 16, 21] that a gyroscope in this orbit precesses about 6,614.4 milli-arcseconds per year around its orbital angular momentum vector (geodetic precession) and about 40.9 milli-arcseconds per year about Earth’s angular momentum vector (frame-dragging)3. Most impressively, GPB should convincingly observe the frame-dragging effect, an effect of the off-diagonal metric elements that originate from the rotation of Earth. Of particular interest to us is that GPB can reach a precision of for the geodetic precession, which as we will see enables precision discrimination4 between GR and a class of torsion theories.

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 Riemann-Cartan 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 world-line 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 Hayashi-Shirafuji 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 Hayashi-Shirafuji theory, and then give the torsion-equivalent of the linearized Kerr solution. In Section 12, we generalize the Hayashi-Shirafuji theory to a two-parameter family of gravity theories, which we will term Einstein-Hayashi-Shirafuji (EHS) theories, interpolating between torsion-free GR and the Hayashi-Shirafuji 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 ground-based 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 Einstein-Hayashi-Shirafuji 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 Riemann-Cartan spacetime

We review the basics of Riemann-Cartan spacetime only briefly here, and refer the interested reader to Hehl et al. [108] for a more comprehensive discussion of spacetime with torsion. Riemann-Cartan spacetime is a connected four-dimensional manifold endowed with metric of Lorentzian signature and an affine connection such that the non-metricity defined by Eq. (11) with respect to the full connection identically vanishes. In other words, the connection in Riemann-Cartan spacetime may have torsion, but it must still be compatible with the metric (). The covariant derivative of a vector is given by

(14)