Abstract
In Chap. 4, we discussed homogeneous world models and introduced the standard model of cosmology. It is based on the cosmological principle, the assumption of a (spatially) homogeneous and isotropic universe. Of course, the assumption of homogeneity is justified only on large scales because observations show us that our Universe is inhomogeneous on small scales—otherwise no galaxies or stars would exist.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Strictly speaking, the cosmic dust cannot be described as a fluid because the matter is assumed to be collisionless. This means that no interactions occur between the particles, except for gravitation. Two flows of such dust can thus penetrate each other. This situation can be compared to that of a fluid whose molecules are interacting by collisions. Through these collisions, the velocity distribution of the molecules will, at each position, assume an approximate Maxwell distribution, with a well-defined average velocity that corresponds to the flow velocity at this point. Such an unambiguous velocity does not exist for dust in general. However, at early times, when deviations from the Hubble flow are still very small, no multiple flows are expected, so that in this case, the velocity field is well defined.
- 2.
This is done by first taking the divergence of (7.13), \(\nabla \cdot \dot{{\boldsymbol u}} = -(\dot{a}/a)\nabla \cdot {\boldsymbol u} - (1/a)\nabla ^{2}\phi =\dot{ a}\delta - (1/a)\nabla ^{2}\phi\), where we made use of (7.12) in the second step. Taking the time derivative of (7.12) yields \(\ddot{\delta }= (\dot{a}/a^{2})\nabla \cdot {\boldsymbol u} - (1/a)\nabla \cdot \dot{{\boldsymbol u}} = -2(\dot{a}/a)\dot{\delta } + (1/a^{2})\nabla ^{2}\phi\). Finally, the Poisson equation is employed.
- 3.
An every-day life example of clustering is the following: the population density of many European countries is of order 100 people per km2, and so the mean separation between two people is on the order of 100 m. For those of you who live in a town or a city, the first morning view from the window shows that typically, you find many people within that distance range, an experience strengthened once you get into your car or use public transport. Obviously, people are highly clustered.
- 4.
This method is indeed used for estimating the galaxy correlation function, although with some important modifications to increase its accuracy and efficiency.
- 5.
This may not look like a ‘standard’ Fourier transform on first sight. However, the relation between P(k) and ξ(r) is given by a three-dimensional Fourier transform. Since the correlation function depends only on the separation \(r = \vert {\boldsymbol r}\vert \), the two integrals over the angular coordinates can be performed explicitly, leading to the form of (7.29).
- 6.
You can convince yourself of this by trying to find another type of function of a scale that does not involve a characteristic length; e.g., sinx does not work if x is a length, since the sine of a length is not defined; one thus needs something like sin(x∕x 0), hence introducing a length-scale. The same arguments apply to other functions, such as the logarithm, the exponential etc. Also note that the sum of two power laws, e.g., Ax α + Bx β defines a characteristic scale, namely that value of x where the two terms become equal.
- 7.
One may wonder why the neutrinos in Fig. 7.7 have a broad distribution in space, and not simply a sharp shell—they all stream out with velocity c. The reason is that the initial conditions were chosen such that they correspond to a growing mode. For those, we argued that any density perturbation is associated with a velocity perturbation. As can be seen from (7.25), the relation between density and peculiar velocity is non-local, i.e., the velocity field associated with a density peak at the center is non-zero at all radii. This velocity field also causes the dark matter distribution to expand—once the neutrinos and the photon-baryon fluid have moved out, the gravitational field inside the dark matter peak is weaker than it would have been predicted from a pure dark matter distribution, yielding an expanding peculiar velocity field near the center.
- 8.
This occurs for the same reason that it takes a stone thrown up into the air the same time to reach its peak altitude as to fall back to the ground from there.
- 9.
This result is obtained from conservation of energy and from the virial theorem. The total energy E tot of the sphere is a constant. At the time of maximum expansion, it is given solely by the gravitational binding energy of the system since then the expansion velocity, and thus the kinetic energy, vanishes. On the other hand, the virial theorem implies that in virial equilibrium \(E_{\mathrm{kin}} = -E_{\mathrm{pot}}/2\), and by combining this with the conservation of energy \(E_{\mathrm{tot}} = E_{\mathrm{kin}} + E_{\mathrm{pot}}\) one is then able to compute E pot in equilibrium and hence the radius and density of the collapsed sphere. For an EdS model, \(r_{\mathrm{vir}} = r_{\mathrm{max}}/2\).
- 10.
Until about 2000, this cluster was the highest-redshift massive cluster known.
- 11.
In practice, the mass of a particle is distributed to all eight neighboring grid points, with the relative proportion of the mass depending on the distance of the particle to each of these grid points.
- 12.
The reason for this is found in the property of the power spectrum of density fluctuations that has been discussed in Sect. 7.5.2, namely that P(k) can be approximated by a power law over a wide range in k. Such a power law features no characteristic scale. For this reason, the properties of halos of high and low mass are scale-invariant, as is clearly visible in Fig. 7.31.
- 13.
The binary pulsar PSR J1915+1606 was discovered in 1974. From the orbital motion of the pulsar and its companion star, gravitational waves are emitted, according to General Relativity. Through this, the system loses kinetic (orbital) energy, so that the size of the orbit decreases over time. Since pulsars represent excellent clocks, and we can measure time with extremely high precision, this change in the orbital motion can be observed with very high accuracy and compared with predictions from General Relativity. The fantastic agreement of theory and observation is considered a definite proof of the existence of gravitational waves. For the discovery of the binary pulsar and the detailed analysis of this system, Russell Hulse and Joseph Taylor were awarded the Nobel Prize in Physics in 1993. In 2003, a double neutron star binary was discovered where pulsed radiation from both components can be observed. This fact, together with the small orbital period of 2.4 h implying a small separation of the two stars, makes this an even better laboratory for studying strong-field gravity.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Schneider, P. (2015). Cosmology II: Inhomogeneities in the Universe. In: Extragalactic Astronomy and Cosmology. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54083-7_7
Download citation
DOI: https://doi.org/10.1007/978-3-642-54083-7_7
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-54082-0
Online ISBN: 978-3-642-54083-7
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)