Finally, we consider an alternative to single-field inflation; in the curvaton scenario the inflaton does not generate all the primordial perturbations. Although this asymmetry cannot be produced by a superhorizon fluctuation in the inflaton field, it can be generated by a superhorizon fluctuation in the curvaton field.
Rocking cosmology’s foundations
Using the Gaussian properties of the density field, Lacey and Cole [14] derived an expression for the probability that a halo of mass M1. We begin in Chapter 2 with an examination of EPS predictions for the merger rate of supermassive black holes and show how using an alternative halo merger rate changes the predicted event rate for the Laser Interferometer Space Antenna (LISA). .
Supermassive black hole merger rates: uncertainties from halo merger theory
In Chapter 2, we show that the two EPS predictions for the LISA event rate from supermassive black hole mergers are almost equal because mergers between haloes of similar mass dominate the event rate. We find that the rate of LISA events from supermassive black hole mergers inferred from EPS theory is thirty percent higher than the alternative merger theory prediction.
Solar system tests of f (R) gravity
However, since a limited range of halo masses dominates the supermassive black hole merger rate, it is possible to find a power-law power spectrum that accurately approximates the VPA prediction for the LISA event rate. Instead, we find that the spacetime in the Solar System is very different from the Schwarzschild solution, implying that 1/Rgravity is ruled out by Solar System gravity tests.
The effects of Chern-Simons gravity on bodies orbiting the Earth
The mathematical results presented in this chapter were derived independently by the author and Tristan Smith and then compared to check for inconsistencies. Mathematical equations were often derived independently by the author and Tristan Smith and then compared to check for deviations.
Superhorizon perturbations and the cosmic microwave background
The first draft of the article was written by Tristan Smith, but revisions were made and additional material was added by the author of this thesis. If the curvaton never dominates the energy density of the Universe, a very large amplitude fluctuation in the curvaton field corresponds to a small fluctuation in the total gravitational potential.
A hemispherical power asymmetry from inflation
If the inflaton, which is not affected by the superhorizon fluctuation in the curvature field, also contributes to the adiabatic power spectrum, then the asymmetry will be diluted on small scales due to the suppression of the isocurvature modes. Throughout the calculation, we present results derived from both versions of the Lacey–Cole clustering rate.
Cosmological event rates
Finally, in Section 2.7 we summarize our results and discuss how these ambiguities in halo merger theory limit our ability to learn about reionization and supermassive black hole formation from LISA observations. Thus, the upper limits of the relevant redshift and SMBH mass intervals are set by the population of supermassive black holes rather than the sensitivity of LISA.
The relationship between halo mass and black hole mass
Assuming that vc =vvir, the halo mass then becomes a redshift-dependent function of the mass of the central black hole:. Therefore, defining Mmin by a minimum haloviral temperature is almost equivalent to defining Mmin by a minimum black hole mass via Eq.
LISA event rates from EPS merger theory
Review of EPS merger theory
The last line of Eq. 2.19) is the source of mass asymmetry in the Lacey-Cole coupling formalism. In fact, the EPS merger theory involves two distinct merger kernels, depending on the order of the mass arguments.
LISA event rates from EPS theory
When Mmin is larger than M∗(z′), the paucity of larger haloes at redshifts higher than z′ leads to a rapid decrease in N(z) as zinc increases beyond z′. The difference between the event rates is always less than a factor of three, so mergers between haloes with mass ratios greater than 1000 cannot make a significant contribution to the event rate.
BKH merger theory
Solving the coagulation equation
For a judicious choice of time variables [τ =−lnδcoll/D(z)], the differentiation of the Press–Schechter mass function introduces noz dependence, and the coagulation equation becomes redshift-invariant. For more complicated spectra, the solidification equation will have to be solved at multiple redshifts.
BKH merger rates for power-law power spectra
However, since the coagulation equation was not solved for the ΛCDM power spectrum, we compare the EPS merger rates with the BKH merger rates for the same power law. The dashed curve is the EPS integration kernel for the power law approximation with n=−2.1 and σ8= 0.98.
Comparison of LISA event rates from BKH and EPS merger theories
The dotted curves are the results derived from the EPS theory for a power law approximation with n=−2.1 andσ8 = 0.98. The dotted curves are the results derived from EPS theory for a power law approximation: n=−2.1 and σ8 = 0.98.
Summary and discussion
The difference between the EPS merger rates for mass ratios in this range is small, so the effect of the EPS mass asymmetry on the LISA event rate due to SMBH mergers is minimal. Testing f(R) gravity in the solar system 1. General relativity predicts that the expansion of the universe should slow down if the universe contains only matter and radiation.
A detailed example: 1/R gravity
The solution for the Solar System is identical to the spacetime derived using the corresponding scalar-tensor theory. We now return to the trace of the field equation, given by Eq. 3.3), and solve for the Ricci scalarR(r) in the presence of the Sun.
The weak-field solution around a spherical star in f (R) gravity
We will now use the expression for R1 given by Eq. 3.37) to solve the field equations for the metric perturbations ν and λ. Transformation of the metric given by Eq. 3.33) to isotropic coordinates, taking a= 1 today, and keeping only terms that are linear in GM/r give.
Case studies in f (R) gravity
The static solution to Eq. Therefore, fRR0 diverges rather than vanishes in the limit where µ→0, and general relativity is not recovered. We note, however, that the static solution to Eq. 3.27) may not describe the current cosmological background in 1/Rn gravity.
Characteristics of viable f (R) theories
Therefore, the scalar field is actually more massive in a region of higher density, provided that the scalar field reaches the minimum of the effective potential in that region (φint). The effective potential is shown for two values of density, with ρint > ρext.
![Figure 3.1: The effective potential of chameleon gravity. The effective potential is the sum of a monotonically decreasing V (φ) (dotted curve) and ρ exp[βφ/m Pl ] (solid curves)](https://thumb-ap.123doks.com/thumbv2/123dok/10405187.0/65.918.253.722.128.557/figure-effective-potential-chameleon-effective-potential-monotonically-decreasing.webp)
Summary and discussion
When these terms are included, the effects of the scalar degree of freedom in (R) gravity leading to deviations from general relativity in the Solar System can be suppressed by the chameleon mechanism [ 133 , 134 ]. During the final preparation of the material presented in this chapter, we learned about recent work in a similar way [138].
Introduction
In Section 4.2, we begin by defining Chern–Simons gravity and deriving the gravitational field equations. In Section 4.3, we discuss the linear theory and derive the gravitomagnetic equations of motion (Chern-Simons Ampere's law).
Chern-Simons gravity
We assume, as in other recent work, that the scalar field associated with the Chern-Simons term varies in time but is spatially homogeneous. In section 4.5 we consider the orbital precession of test bodies in this spacetime, as well as the orbital precession of gyroscopes, and determine the regions of the Chern-Simons gravity parameter space explored with the LAGEOS and Gravity Probe B satellites. .
The Chern-Simons gravitomagnetic equations
Since in the following we will only require gravitomagnetic fields, we will be primarily interested in the time-space components of the equations of the linearized field. In general relativity, the space-time components of the linearized field equations have the form
Gravitomagnetism due to a spinning sphere in Chern-Simons gravity
Calculation of the vector potential
So far, we have not discussed any boundary conditions on the gravitomagnetic field B~ at the surface of the sphere. Getting the equalizer curl. 4.36) confirms that ∇ ×~ A~GR is continuous along the surface of the sphere.
The gravitomagnetic field
In Chern-Simons gravity, the gravitomagnetic field is the sum of B~GRandB~CS, and the solid curves are the components of B~CS. 4.1, the Chern-Simons gravitomagnetic field has a non-zero azimuthal component Bφ if we take ~ω to lie in the ˆz direction.
Orbital and Gyroscopic precession
Orbital precession
The length of the ascending node (Ω) is the angle between the X axis and the nodal line segment connecting the origin of the coordinate system to the point where the orbit crosses the XY plane as it travels in the positive Z direction. A 1% Lense-Thirring drag check will improve this limited nmc by a factor of approximately five.
Gyroscopic precession
We conclude that our MCS constraints are not sensitive to the details of the Earth's density profile.
Summary and discussion
Most recently, an analysis of the WMAP first-year data [158] found that our nearly homogeneous patch of the Universe extends up to 3900 times the cosmological horizon [162] . Thus, the strongest constraints on the amplitude and wavelength of a single superhorizon mode arise from measurements of the CMB quadrupole and octupole.
The Grishchuk-Zel’dovich effect: A brief review
In the next chapter, we will investigate how a superhorizon disturbance during slow-roll inflation can generate an anomalous feature of the CMB: the fluctuation amplitude on large scales (ℓ ~<64) is 7% larger on one side of the sky than on the other side [ 46]. We will find that a superhorizon perturbation in the curvaton field can generate the observed power asymmetry without inducing prohibitively large CMB anisotropies.
CMB anisotropies from superhorizon potential perturbations
The combination of the SW effect, the ISW effect and the Doppler effect gives the total CMB temperature anisotropy given by a potential perturbation of the form given in Eq. However, if the superhorizon perturbation is a single mode of the form Ψ = Ψ~ksin(~k·~x+̟), then the requirement that Ψ∼<1 everywhere, even outside our Hubble volume, leads to an additional constraint: Ψ ~k(τdec)∼<1 implies that (kxdec)∼>∆Ψ/cos̟.
Application to curvaton perturbations
The solid curve is the limit imposed by the CMB quadrupole, and it vanishes for ̟σ = π/6 and ̟σ = 5π/6. The solid curve is the boundary of the CMB quadrupole, and the long-dashed curves are boundaries from the CMB octupole.
Summary and discussion
This hemispheric power asymmetry can be parameterized as a dipolar modulation of the temperature anisotropy field [44, 46]; the temperature fluctuations in the ˆ. In section 6.4 we investigate how a hemispheric power asymmetry can be created by a superhorizon fluctuation in the curvaton field in two limiting cases of the curvaton scenario.
A scale-invariant power asymmetry
Single-field models
The strength asymmetry can be diluted on smaller scales by introducing discontinuities in the inflaton potential and its derivative, which change the relative contributions of the curvaton and the inflaton field to the original perturbations [165] . The observed 7.2% variation in the amplitude of CMB temperature fluctuations [46] corresponds to the power asymmetry ∆PΨ/PΨ≃2A= 0.144, where A is defined by Eq.
The curvaton model
The lower bound for ξ arises from the requirement that the fractional change in the curvaton field in the observable universe must be less than unity. The superhorizon fluctuation in the curvaton field could also be a superhorizon inhomogeneity that is not completely erased by inflation.
Review of isocurvature perturbations
Isocurvature perturbations in the curvaton scenario
After curvaton decay, there are superhorizon adiabatic perturbations ζ(f) and superhorizon dark matter isocurvature perturbations Smγ. In the opposite limit, in which the curvaton contribution to the dark matter density is negligible, we have
Isocurvature modes in the cosmic microwave background
Since their bounds on α and γ are marginalized over a range of values unattainable in the curvaton scenario, these constraints are not applicable to our model. Recall from section 6.2.2 that the curvaton also introduces non-Gaussianity in adiabatic perturbations; The fNL for mixed inflaton and curvaton perturbations is given by.
A power asymmetry from curvaton isocurvature
Case 1: The curvaton creates most of the dark matter
If most of the dark matter is formed when the curvaton decays, then TSS ≃1−R, as in Eq. where we have retained only the leading term in R in the coefficients of ˆCℓiso and ˆCℓcor. We conclude that the curvaton cannot generate the observed power asymmetry if the dark matter arises during curvaton decay.
Case 2: The curvaton’s contribution to the dark matter is negligible
The shaded area in the upper right corner is excluded by the upper limit of the isocurvature power (α <0.072), and the left shaded area is excluded by the invariance of the resulting asymmetry scale ( ˜A/ξ < 6). We see that only a limited region of the κ-ξ plane can give rise to asymmetries with A≥0.072 while satisfying the isocurvature power upper limit (α <0.072) and the quasar constraint ( ˜A/ξ >6).
Summary and discussion
We can therefore suppress any observable signature of the superhorizon curvature fluctuation in the CMB without changing the asymmetry by reducing the energy density of the curvature. We then expand to linear order in the perturbationϕ1, and write Eq. A.4) in terms of the physical metric µν.
Dipole cancellation in a universe with an exotic fluid
Relating halo masses to supermassive black hole masses
The critical halo mass M ∗ (z)
The two EPS merger kernels
The rate of SMBH mergers per comoving volume for different minimum halo masses. 20
The gravitational-wave event rate from SMBH mergers as a function of the maximum
The halo mass range that dominates the rate of SMBH mergers per comoving volume. 25
Equal-mass BKH merger kernels for z = 0
Equal-mass BKH merger kernels for z = 5
The two EPS merger kernels and the BKH merger kernel
The rate of SMBH mergers per comoving volume from BKH and EPS merger rates. 34
The effective potential of chameleon gravity
The gravitomagnetic field generated by a rotating sphere in Chern-Simons gravity
An orbital diagram showing the longitude of the ascending node
The Lense-Thirring drag for the LAGEOS satellites in Chern-Simons gravity
The gyroscopic precession for Gravity Probe B in Chern-Simons gravity
The evolution of the gravitational potential Ψ in a ΛCDM Universe with radiation
Contributions to the CMB dipole from a superhorizon perturbation
