Combining general relativity and quantum theory: points of conflict and contact

The issues related to bringing together the principles of general relativity and quantum theory are discussed. After briefly summarising the points of conflict between the two formalisms I focus on four specific themes in which some contact has been established in the past between GR and quantum field theory: (i) The role of planck length in the microstructure of spacetime (ii) The role of quantum effects in cosmology and origin of the universe (iii) The thermodynamics of spacetimes with horizons and especially the concept of entropy related to spacetime geometry (iv) The problem of the cosmological constant.Comment: Invited Talk at "The Early Universe and Cosmological Observations: a Critical Review", UCT, Cape Town, 23-25 July,2001; to appear in Class.Quan.Gra

Nonlinear evolution of density perturbations using approximate constancy of gravitational potential

During the evolution of density inhomogeneties in an $\Omega=1$, matter dominated universe, the typical density contrast changes from $\delta\simeq 10^{-4}$ to $\delta\simeq 10^2$. However, during the same time, the typical value of the gravitational potential generated by the perturbations changes only by a factor of order unity. This significant fact can be exploited to provide a new, powerful, approximation scheme for studying the formation of nonlinear structures in the universe. This scheme, discussed in this paper, evolves the initial perturbation using a Newtonian gravitational potential frozen in time. We carry out this procedure for different intial spectra and compare the results with the Zeldovich approximation and the frozen flow approximation (proposed by Mattarrese et al. recently). Our results are in far better agreement with the N-body simulations than the Zeldovich approximation. It also provides a dynamical explanation for the fact that pancakes remain thin during the evolution. While there is some superficial similarity between the frozen flow results and ours, they differ considerably in the velocity information. Actual shell crossing does occur in our approximation; also there is motion of particles along the pancakes leading to further clumping. These features are quite different from those in frozen flow model. We also discuss the evolution of the two-point correlation function in various approximations.Comment: 10 pages, TeX, 6 figures available on request, IUCAA -14/93( Corrections for mailing error

A New Statistical Indicator to Study Nonlinear Gravitational Clustering and Structure Formation

In an expanding universe, velocity field and gravitational force field are proportional to each other in the linear regime. Neither of these quantities evolve in time and these can be scaled suitably so that the constant of proportionality is unity and velocity and force field are equal. The Zeldovich approximation extends this feature beyond the linear regime, until formation of pancakes. Nonlinear clustering which takes place {\it after} the breakdown of Zeldovich approximation, breaks this relation and the mismatch between these two vectors increases as the evolution proceeds. We suggest that the difference of these two vectors could form the basis for a powerful, new, statistical indicator of nonlinear clustering. We define an indicator called velocity contrast, study its behaviour using N-Body simulations and show that it can be used effectively to delineate the regions where nonlinear clustering has taken place. We discuss several features of this statistical indicator and provide simple analytic models to understand its behaviour. Particles with velocity contrast higher than a threshold have a correlation function which is biased with respect to the original sample. This bias factor is scale dependent and tends to unity at large scales.Comment: 12 pages, 8 figures, LaTeX with uuencoded figures, uses MN.sty and epsf.sty; Discussion has been enlarged to clarify a few points. Introduction has been added. Some figures have change

Critical Index and Fixed Point in the Transfer of Power in Nonlinear Gravitational Clustering

We investigate the transfer of power between different scales and coupling of modes during non-linear evolution of gravitational clustering in an expanding universe. We start with a power spectrum of density fluctuations that is exponentially damped outside a narrow range of scales and use numerical simulations to study evolution of this power spectrum. Non-Linear effects generate power at other scales with most power flowing from larger to smaller scales. The ``cascade'' of power leads to equipartition of energy at smaller scales, implying a power spectrum with index $n\approx -1$. We find that such a spectrum is produced in the range $1 < \delta < 200$ for density contrast $\delta$. This result continues to hold even when small scale power is added to the initial power spectrum. Semi-analytic models for gravitational clustering suggest a tendency for the effective index to move towards a critical index $n_c\approx -1$ in this range. For n<n_c, power in this range grows faster than linear rate, while if n>n_c, it grows at a slower rate - thereby changing the index closer to n_c. At scales larger than the narrow range of scales with initial power, a k^4 tail is produced. We demonstrate that non-linear small scales do not effect the growth of perturbations at larger scales.Comment: Title changed. Added two figures and some discussion. Postscript file containing all the figures is available at http://www.ast.cam.ac.uk/~jasjeet/papers/powspec.ps.gz Accepted for publication in the MNRA

Cohort profile of the UK Biobank: diagnosis and characteristics of cerebrovascular disease

Purpose: The UK Biobank is a large-scale biomedical resource, containing sociodemographic and medical information, including data on a previous diagnosis of stroke or transient ischaemic attack (TIA). We described these participants and their medication usage. Participants: We identified participants who either self-reported or were identified from a nurse-led interview, having suffered a stroke or a TIA and compared them against participants without stroke ort TIA. We assessed their risk factor burden (sex, age, deprivation, waist to hip ratio (WHR), hypertension, smoking, alcohol intake, diabetes, physical exercise and oral contraception use (oral contraceptive pill, OCP)) and medication usage. Findings: to date We studied 502 650 people (54.41% women), 6669 (1.23%) participants self-reported a stroke. The nurse-led interview identified 7669 (1.53%) people with stroke and 1781 (0.35%) with TIA. Hypertension, smoking, higher WHR, lower alcohol consumption and diabetes were all more common in people with cerebrovascular disease (p&lt;0.0001 for each). Women with cerebrovascular disease were less likely to have taken the OCP (p=0.0002). People with cerebrovascular disease did more exercise (p=0.03). Antithrombotic medication was taken by 81% of people with stroke (both self-report and nurse-led responders) and 89% with TIA. For self-reported stroke, 63% were taking antithrombotic and cholesterol medications, 54% taking antithrombotic and antihypertensive medications and 46% taking all 3. For the nurse-led interview and TIA, these figures were 65%, 54% and 46%, and 70%, 53% and 45%, respectively. Future plans: The UK Biobank provides a large, generalisable and contemporary data source in a young population. The characterisation of the UK Biobank cohort with cerebrovascular disease will form the basis for ongoing research using this data source

Vacuum Fluctuations of Energy Density can lead to the observed Cosmological Constant

The energy density associated with Planck length is $\rho_{uv}\propto L_P^{-4}$ while the energy density associated with the Hubble length is $\rho_{ir}\propto L_H^{-4}$ where $L_H=1/H$. The observed value of the dark energy density is quite different from {\it either} of these and is close to the geometric mean of the two: $\rho_{vac}\simeq \sqrt{\rho_{uv} \rho_{ir}}$. It is argued that classical gravity is actually a probe of the vacuum {\it fluctuations} of energy density, rather than the energy density itself. While the globally defined ground state, being an eigenstate of Hamiltonian, will not have any fluctuations, the ground state energy in the finite region of space bounded by the cosmic horizon will exhibit fluctuations $\Delta\rho_{\rm vac}(L_P, L_H)$. When used as a source of gravity, this $\Delta \rho$ should lead to a spacetime with a horizon size $L_H$. This bootstrapping condition leads naturally to an effective dark energy density $\Delta\rho\propto (L_{uv}L_H)^{-2}\propto H^2/G$ which is precisely the observed value. The model requires, either (i) a stochastic fluctuations of vacuum energy which is correlated over about a Hubble time or (ii) a semi- anthropic interpretation. The implications are discussed.Comment: r pages; revtex; comments welcom

Surface Density of Spacetime Degrees of Freedom from Equipartition Law in theories of Gravity

I show that the principle of equipartition, applied to area elements of a surface which are in equilibrium at the local Davies-Unruh temperature, allows one to determine the surface number density of the microscopic spacetime degrees of freedom in any diffeomorphism invariant theory of gravity. The entropy associated with these degrees of freedom matches with the Wald entropy for the theory. This result also allows one to attribute an entropy density to the spacetime in a natural manner. The field equations of the theory can then be obtained by extremising this entropy. Moreover, when the microscopic degrees of freedom are in local thermal equilibrium, the spacetime entropy of a bulk region resides on its boundary.Comment: v1: 20 pages; no figures. v2: Sec 4 added; 23 page