Category:Deceleration Parameter

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Problem 1

problem id: 1301_1

Show that for a spatially flat Universe consisting of one component with equation of state $p=w\rho$ the deceleration parameter is equal to $q=(1+3w)/2$.


Problem 2

problem id: 1301_2

Find generalization of the relation $q=(1+3w)/2$ for the non-flat case.


Problem 3

problem id: 1301_3

Find deceleration parameter for multi-component non-flat Universe.


Problem 4

problem id: 1301_4

Show that the expression for deceleration parameter obtained in the previous problem can be presented in the following form \[q=\frac{\Omega_{total}}2+\frac32\sum\limits_i w_i \Omega_i.\]


Problem 5

problem id:

Let us consider the model of two-component Universe [ J. Ponce de Leon, cosmological model with variable equations of state for matter and dark energy, arXiv:1204.0589]. Such approximation is sufficient to achieve good accuracy on each stage of its evolution. At the present time the two dark components - dark matter and dark energy - are considered to be dominating. We neglect meanwhile the interaction between the components and as a result they separately satisfy the conservation equation. Let us assume that the state equation parameter for each component depends on the scale factor \begin{align} \nonumber p_{de} & = W(a) \rho_{de};\\ \nonumber p_m & = w(a)\rho_{m}. \end{align} Express the relative densities $\Omega_m$ and $\Omega_{de}$ in terms of the deceleration parameter and the state equation parameters $w(a)$ and $W(a)$.


Problem 6

problem id: 1301_6

Find the upper and lower limits on the deceleration parameter using results of the previous problem.


Problem 7

problem id: 1301_7

Find the relation between the total pressure and the deceleration parameter for the flat one-component Universe


Problem 8

problem id: 1301_8

The expansion of pressure by the cosmic time is given by \[p(t)=\left.\sum\limits_{k=0}^\infty\frac1{k!}\frac{d^kp}{dt^k}\right|_{t=t_0}(t-t_0)^k.\] Using cosmography parameters , evaluate the derivatives up to the fourth order.


Problem 9

problem id: 1301_9

Show that for one-component flat Universe filled with ideal fluid of density $\rho$.


Problem 10

problem id: 1301_10

For what values of the state parameter $w$ the rate of expansion of a one-component flat Universe increases with time?


Problem 11

problem id: 1301_11

Express the age of the spatially flat Universe filled with a single component with equation of state $p=w\rho$ through the deceleration parameter.


Problem 12

problem id: 1301_12

Suppose we know the current values of the Hubble constant $H_0$ and the deceleration parameter $q_0$ for a closed Universe filled with dust only. How many times larger will it ever become? Find lifetime of such a Universe.


Problem 13

problem id: 1301_13

In a closed Universe filled with non-relativistic matter the current values of the Hubble constant is $H_0$, the deceleration parameter is $q_0$. Find the current age of this Universe.


Problem 14

problem id: 1301_14

In a closed Universe filled with dust the current value of the Hubble constant is $H_0$ and of the deceleration parameter $q_0$.

a) What is the total proper volume of the Universe at present time?

b) What is the total current proper volume of space occupied by matter which we are presently observing?

c) What is the total proper volume of space which we are directly observing?


Problem 15

problem id: 1301_15

For the closed ($k=+1$) model of Universe, filled with non-relativistic matter, show that solutions of the Friedmann equations can be represented in terms of the two parameters $H_0$ and $q_0$. [Y.Shtanov, Lecture Notes on theoretical cosmology, 2010 ]


Problem 16

problem id: 1301_16

Do the same as in the previous problem for the case of open ($k=-1$) model of Universe.


Classification of models of Universe based on the deceleration parameter

Problem 17

problem id: 1301_17

When the rate of expansion never changes, and $\dot a$ is constant, the scaling factor is proportional to time $t$, and the deceleration term is zero. When the Hubble parameter is constant, the deceleration parameter $q$ is also constant and equal to $-1$, as in the de Sitter model. All models can be characterized by whether they expand or contract, and accelerate or decelerate. Build a classification of such type using the signature of the Hubble parameter and the deceleration parameter.


Problem 18

problem id: 1301_18

Can the Universe variate its type of evolutions in frames of the classification given in the previous problem?


Problem 19

problem id: 1301_19

Point out possible regime of expansion of Universe in the case of constant deceleration parameter.


Problem 20

problem id: 1301_20

Having fixed the material content we can classify the model of Universe using the connection between the deceleration parameter and the spatial geometry. Perform this procedure in the case of Universe filled with non-relativistic matter.


Problem 21

problem id: 1301_21

Models of the Universe can be classified basing on the relation between the deceleration parameter and age of the Universe. Build such a classification in the case of Universe filled with non-relativistic matter (the Einstein-de Sitter Universe)


Problem 22

problem id: 1301_22

Show that sign of the deceleration parameter determines the difference between the actual age of the Universe and the Hubble time.


Problem 23

problem id: 1301_23

Suppose the flat Universe is filled with non-relativistic matter with density $\rho_m$ and some substance with equation of state $p_X=w\rho_X$. Express the deceleration parameter through the ratio $r\equiv\rho_m/\rho_X$.


Problem 24

problem id: 1301_23_1

Obtain Friedmann equations for the case of spatially flat $n$-dimensional Universe. (see Shouxin Chen, Gary W. Gibbons, Friedmann's Equations in All Dimensions and Chebyshev's Theorem, arXiv: 1409.3352)


Problem 25

problem id: 1301_23_2

Analyze exact solutions of the Friedmann equations obtained in the previous problem in the case of flat ($k=0$) $n$-dimensional Universe filled with a barotropic liquid with the state equation \begin{equation} \label{10} p_m=w \rho_m. \end{equation} and obtain corresponding explicit expressions for the deceleration parameter.

Deceleration as a cosmographic parameter

Problem 26

problem id: 1301_24

Make transition from the derivatives w.r.t. cosmological time to that w.r.t. conformal time in definitions of the Hubble parameter and the deceleration parameter.


Problem 27

problem id: 1301_25

Make Taylor expansion of the scale factor in time using the cosmographic parameters.


Problem 28

problem id: 1301_26

Make Taylor expansion of the redshift in time using the cosmographic parameters.


Problem 29

problem id: 1301_27

Show that \[q(t)=\frac{d}{dt}\left(\frac1H\right)-1.\]


Problem 30

problem id: 1301_28

Show that the deceleration parameter as a function of the red shift satisfies the following relations \begin{align} \nonumber q(z)=&\frac{1+z}{H}\frac{d H}{dz}-1;\\ \nonumber q(z)=&\frac12(1+z)\frac1{H^2}\frac{d H^2}{dz}-1;\\ \nonumber q(z)=&\frac12\frac{d\ln H^2}{d\ln(1+z)}-1;\\ \nonumber q(z)=&\frac{d\ln H}{dz}(1+z)-1. \end{align}


Problem 31

problem id: 1301_29

Show that the deceleration parameter as a function of the scale factor satisfies the following relations \begin{align} \nonumber q(a)=&-\left(1+\frac{a\frac{dH}{da}}{H}\right);\\ \nonumber q(a)=&\frac{d\ln(aH)}{d\ln a}. \end{align}


Problem 32

problem id: 1301_30

Show that the derivatives $dH/dz$, $d^2H/dz^2$, $d^3H/dz^3$ and $d^4H/dz^4$ can be expressed through the deceleration parameter $q$ and other cosmological parameters.


Problem 33

problem id: 1301_31

Use results of the previous problem to make Taylor expansion of the Hubble parameter in redshift.


Problem 34

problem id: 1301_32

Express derivatives of the Hubble parameter squared w.r.t. the redshift $d^iH^2/dz^i$, $i=1,2,3,4$ in terms of the cosmographic parameters.


Problem 35

problem id: 1301_33

Express the current values of deceleration and jerk parameters in terms of $N=-\ln(1+z)$.


Problem 36

problem id: 1301_34

Express time derivatives of the Hubble parameter in terms of the cosmographic parameters.


Problem 37

problem id: 1301_35

Express the deceleration parameter as power series in redshift $z$ or $y$-redshift $z/(1+z)$.


Problem 38

problem id: 1301_36

Show that the Hubble parameter is connected to the deceleration parameter by the integral relation \[H=H_0\exp\left[\int\limits_0^z[q(z')+1]d\ln (1+z')\right].\]


Problem 39

problem id: 1301_37

Show that derivatives of the lower order parameters can be expressed through the higher ones, for instance \[\frac{dq}{d\ln(1+z)}=j-q(2q+1).\]


Cosmological scalars and the Friedmann equation

Dunajski and Gibbons [M. Dunajski, Gary Gibbons, Cosmic Jerk, Snap and Beyond, arXiv:0807.0207] proposed an original way to test the General Relativity and the cosmological models based on it. The procedure implies expressing the Friedmann equation in terms of directly measurable cosmological scalars constructed out of higher derivatives of the scale factor, i.e. cosmographic parameters $H,q,j,s,l$. In other words, the key idea is to treat the Friedmann equations as one algebraic constraint between the scalars. This links the measurement of the cosmological parameters to a test of General Relativity, or any of its modifications.


Problem 40

problem id: 1301_38

Express the curvature parameter $k$ terms of the cosmographic parameters for the case of Universe filled with non-interacting cosmological constant and non-relativistic matter.


Problem 41

problem id: 1301_39

Do the same as in the previous problem for the case of Universe filled with non-interacting non-relativistic matter $\rho_m=M_m/a^3$ and radiation $\rho_r=M_r/a^4$.


Problem 42

problem id: 1301_40

Check the expressions for the curvature $k$ obtained in the previous problem for two cases: a) a flat Universe solely filled with non-relativistic matter; b) a flat Universe solely filled with radiation.


Problem 43

problem id: 1301_41

Find relation between the cosmographic parameters free of any cosmological parameter for the case of Universe considered in the problem #1301_38.


Problem 44

problem id: 1301_42

Perform the same procedure for the Chaplygin gas with the equation of state $p=-A/\rho$.


Problem 45

problem id: 1301_43

Perform the same procedure for the generalized Chaplygin gas with the equation of state $\rho=-A/\rho^\alpha$.

Averaging deceleration parameter

Since the deceleration parameter $q$ is a slowly varying quantity (e.g. $q = 1/2$ for matter-dominated case and $q = -1$ in the Universe dominated by dark energy in form of cosmological constant), then the useful information is contained in its time average value, which is very interesting to obtain without integration of the equations of motions for the scale factor. Let us see how it is possible [J.Lima, Age of the Universe, Average Deceleration Parameter and Possible Implications for the End of Cosmology, arXiv:0708.3414]. For that purpose let us define average value $\bar q$ of this parameter on time interval $[0,t_0]$ with the expression \[\bar q(t_0)=\frac1{t_0}\int\limits_0^{t_0}q(t)dt.\]


Problem 46

problem id: 1301_44

Express current value of the average deceleration parameter in terms of the Hubble parameter.


Problem 47

problem id: 1301_45

Show that current age of the Universe depends solely on average value of the deceleration parameter.


Problem 48

problem id: 1301_46

Show that the Hubble time $H_0)^{-1}$ represents a characteristic time scale for age of the Universe at any stage of the evolution.


Energy conditions in terms of the deceleration parameter

Dynamic model-independent constraints on the kinematics of the Universe can further be obtained from the so-called energy conditions. These conditions, based on quite general physical principles, impose restrictions on the components of the energy-momentum tensor $T_{\mu\nu}$. In choosing a model for the medium (a model, but not the equation of state!), these conditions can be transformed into inequalities restricting the possible values of pressure and density of the medium. In terms of density and pressure the energy conditions take on the form \begin{align} \nonumber NEC&\Rightarrow & \rho+p\ge&0, & &\\ \nonumber WEC&\Rightarrow & \rho\ge&0,&\rho+p\ge&0,\\ \nonumber SEC&\Rightarrow & \rho+3p\ge&0,&\rho+p\ge&0,\\ \nonumber DEC&\Rightarrow & \rho\ge&0,&-\rho\le p\le&\rho. \end{align} Here, NEC, WEC, SEC, and DEC correspond to the zero, weak, strong, and dominant energy conditions. Because these conditions do not require any definite equation of state for the substance filling the Universe, they impose very simple and model-independent constraints on the behavior of the energy density and pressure. Hence, the energy conditions provide one of the possibilities for explaining the evolution of the Universe on the basis of quite general principles.


Problem 49

problem id: 1301_47

Express the energy conditions in terms of the scale factor and its derivatives.


Problem 50

problem id: 1301_48

Transform the energy conditions to constraints on the deceleration parameter for the flat Universe.


Problem 51

problem id: 1301_49

Analyze the constraints on the regimes of accelerated and decelerated expansion of the Universe following from the energy conditions.


Distance-Deceleration Parameter Relations

In cosmology there are many different and equally natural definitions of the notion of distance between two objects or events. In particular, the luminosity distance $d_L$ of an object at redshift $z$ is $d_L=(L/2\pi F)^{1/2}$, where $L$ is the bolometric luminosity for a given object and $F$ is the bolometric energy flux received from that object. The expression for the luminosity distance in a FLRW Universe is \begin{equation}\label{distance_7_1} d_L(z)=(1+z)\left\{ \begin{array}{lr} R\sinh\left[\frac1{H_0R}\int\limits_0^z\frac{dz'}{E(z')}\right], & open\\ H_0^{-1}\int\limits_0^z\frac{dz'}{E(z')}, & flat\\ R\sin\left[\frac1{H_0R}\int\limits_0^z\frac{dz'}{E(z')}\right], & closed \end{array} \right.\end{equation} (Here and below we set $c=1$). Here $R$ is the (comoving) radius of curvature of the open or closed Universe, $E=H/H_0$.



Problem 52

problem id: 1301_50

Represent the luminosity distance in the flat Universe in terms of the deceleration parameter.


Problem 53

problem id: 1301_51

Find expression for the luminosity distance up to terms of order of $z^2$.


Problem 54

problem id: 1301_52

Find expression for the luminosity distance in the next order $O(z^3)$ in the redshift.


Problem 55

problem id: 1301_53

Calculate the luminosity distance in the Universe filled with non-relativistic matter (Einstein-de Sitter model ($k=0$)).


Problem 56

problem id: 1301_54

The expression (\ref{distance_7_1}) allows to find the luminosity distance given the function $H(z)$. Solve the inverse problem for the flat case: find the Hubble parameter as a function of the luminosity distance [S. Nesseris and J. Garcia-Bellido, Comparative analysis of model-independent methods for exploring the nature of dark energy, arXiv:1306.4885].


Problem 57

problem id: 1301_55

Use result of the previous problem to express deceleration parameter a s a function of luminosity distance and its derivatives.


Horizons

Problem 58

problem id: 1301_56

The Hubble sphere in general does not coincide with the horizons, except when it becomes degenerate with the particle horizon at $q=1$ and with event horizon at $q=-1$. Show that.


Problem 59

problem id: 1301_57

Show that the Hubble sphere contracts when $q<-1$, remains stationary when $q=-1$ and expands when $q>-1$.


Problem 60

problem id: 1301_58

Analyze correspondence between kinematics of the Hubble sphere and boundaries of the observable Universe for different expansion regimes.


Problem 61

problem id: 1301_59

Calculate the derivatives $dL_p/dt$ and $dL_e/dt$.


Problem 62

problem id: 1301_60

Calculate the derivatives $d^2L_p/dt^2$ and $d^2L_e/dt^2$.


Problem 63

problem id: 1301_61

Find the expressions for the current particle horizon in the single-component Universe filled with non-relativistic matter.


Power-Law Universes

Problem 64

problem id: 1301_62

Assume that the scale factor $a(t)$ varies as $t^n$, where $t$ is the age of Universe and $n=const$. Show that in decelerated expanding Universe $n<1$.


Problem 65

problem id: 1301_63

Show that if the scale factor $a(t)$ varies as $t^n$, \[L_p=\frac{R_h}{q}.\]


Problem 66

problem id: 1301_64

How, in a Universe of age $t$ can causally connected distances of $L\gg ct$ exist?


The Effects of a Local Expansion of the Universe

Problem 67

problem id: 1301_65

Considering the radial motion of a test particle in a spatially-flat expanding Universe find the Newtonian limit the radial force $F$ per unit mass at a distance $R$ from a point mass $m$.


Problem 68

problem id: 1301_66

Find the Newtonian limit of the radial force $F$ per unit mass at a distance $R$ from a point mass $m$ in a Universe which contains no matter (or radiation), but only dark energy in the form of a non-zero cosmological constant $\Lambda$.


Problem 69

problem id: 1301_67

Solve the previous problem for the case finite (i.e. non-pointlike) spherically-symmetric massive objects.


Problem 70

problem id: 1301_68

Show that a non-zero $\Lambda$ should set a maximum size, dependent on mass, for galaxies and clusters.


Problem 71

problem id: 1301_69

The radius $R=R_F$ (see previous problem) does not necessarily correspond to the maximum possible size of the galaxy or cluster: many of the gravitationally-bound particles inside $R_F$ may be in unstable circular orbits. Therefore the so-called "outer" radius becomes of great importance, which one may interpret as the maximum size of the object, that is corresponding to the largest stable circular orbit $R_S$. Find in the Newtonian limit radius of the largest stable circular orbit $R_S$ in the case of gravity field created by a point mass $m$.


Problem 72

problem id: 1301_70

If we consider $R_S(t)$ (see previous problem) as the maximum possible size of a massive object at cosmic time $t$, and assume that the object is spherically-symmetric and has constant density, then it follows that there exists a time-dependent minimum density (due to the maximum size). Determine the corresponding density.


Problem 73

problem id: 1301_71

Show that minimum relative density $\Omega_{min}\equiv\rho_{min}/\rho_{crit}$ is determined only by the deceleration parameter.


Problem 74

problem id: 1301_72

An important characteristics -- the minimum fractional density contrast $\delta_{min}(t)\equiv[\rho_{min}(t)-\rho_m(t)]/\rho_m(t)$ is immediately related to the deceleration parameter. Estimate this quantity in the SCM.

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