−*μ*|*y*| ,

*n*(*y*, *t*) � *e*

−2*μ*|*y*|

(*R* − Λ(5))

+ 1 2

˜

The action *I* is invariant under the gauge transformations

**6. Period of late cosmology**

where we used that *μM*2

can be approximated as

and the approximate 5D metric is

quadratic in fluctuations is

*I* = 1 2 *d*5*x* −*g*(5)

subsequent simplifications.

the point |*y*¯|

Universe condition of late cosmology is

*ρr*(*T*)

*μ* = *κ*2*ρ*ˆ*r*(*T*)

*ds*2 � *dy*2 + *e*

**7. Fluctuations of the background metric**

$$
\mu m\_{\rm tr} \simeq \mu e^{-\mu \mathfrak{g}} \left( n \pi + \frac{\pi}{2} \right) \tag{60}
$$

and the normalized eigenmode *hm*(0)

$$h\_m(0) \simeq (\mu e^{-\mu \tilde{y}})^{1/2}.\tag{61}$$

For the following we need the sum ∑*n h*2 *mn* (0), where *mn* is determined by (60). Because of a narrow spacing between the levels, we change summation to integration and obtain

$$\sum\_{n} h\_{m\_{\pi}}^{2}(0) \simeq \int \frac{dm \, e^{\mu g}}{\mu \pi} \mu e^{-\mu g} = \int \frac{dm}{\pi} . \tag{62}$$

The integral (62) is independent of *y*¯. The same measure of integration was obtained in Langlois; Sorbo & Rodriguez-Martinez (2002), where the authors used the graviton modes of the Randal-Sundrum II model (Randall & Sundrum, 1999) without matter, in which case the integration over *y* extends to infinity and the spectrum is continuous. Similarity of the results can be traced to the fact that we performed calculations in the period of late cosmology neglecting the terms of order *O*(*ρ*/*μ*) as compared to unity.

#### **7.1 Equations for eigenmodes**

In the background of the approximate metric (56), in the region 0 < *y* < *y*¯, in the gauge *Dμhμν*(*y*, *x*) = 0, *h μ μ*(*y*, *x*) = 0, the (*μν*) components of the field equations for fluctuations are

$$h^{\prime\prime}\_{\mu\nu} - 4\mu^2 h\_{\mu\nu} + b^{-1}(y)D\_{\rho}D^{\rho}h\_{\mu\nu} + \delta(y)4\mu h\_{\mu\nu} = 0. \tag{63}$$

We expand the functions *h μ ν* (*x*, *y*) as

$$h\_{\nu}^{\mu}(\mathfrak{x}, \mathfrak{y}) = \sum\_{m} \phi\_{(m)\nu}^{\mu}(\mathfrak{x}) h\_{m}(\mathfrak{y})\_{\nu}$$

where the functions *h*< *m*(*y*) satisfy the equation 5

$$
\hbar l\_m''(y) - 4\mu^2 \hbar\_m(y) + e^{2\mu|y|} m\_<^2 \hbar\_m(y) + \delta(y) 4\mu \hbar\_m(y) = 0. \tag{64}
$$

5 Wherever it does not lead to ambiguity, we omit the (sub)superscripts < and >.

At temperatures *T* � *μ* at which production of gravitons is sufficiently intensive, the

Graviton Emission in the Bulk and Nucleosynthesis in a Model with Extra Dimension 355

∼ *T μ g*1/2

*μy*¯/2 cos(*m*˜(*e*

*m*(*y*) and *h*>

1/2 *e*

√*m*˜ > *m*˜ <

∗ (*T*) *T*2 *μMpl*

*μy*¯ − 1)). (73)

*m*(*y*), we sew the envelopes of their

*μy*¯/2 (74)

1/2

*me*˜ −*μy*¯ ∼ *T*

*m*> (*y*) = *C*˜

2*C πm*˜< *e*

norm (69), we obtain the normalized eigenmode *hm*(0) (61).

*I* = *κ d*4*x*

 *d*3*k*1 (2*π*)32*E*1

Here *f i* are the Bose/Fermi distributions of colliding particles and *Mi*

*h*>

Instead of sewing the oscillating functions *h*<

**8. Production of Kaluza-Klein gravitons**

with the interaction Lagrangian

Kolb & Turner, 1990)

*dt* + 4*Hρ*ˆ = − ∑*n* ∑

*i*

 *d*3 *p* (2*π*)3

*dρ*ˆ

is large and we can substitute the asymptotics of the Bessel functions

*μ ρ μ*

> 1 2 *πm*˜ *e*

−*μy*¯/2 = *C*˜

1 = *Ce*−*μy*¯

*C*˜

1 2 *πm*˜>

The norm (72) is smaller than (69). Effectively, we neglect the contribution from the region *y* > *y*¯ and impose the condition *hm*(*y*¯) = 0, which yields us the spectrum (60). Using the

In this section we calculate the rate of production of Kaluza-Klein gravitons in interactions of particles of the hot matter on the brane in the radiation-dominated period. The leading contribution to this process is given by the annihilation reactions *ψi* + *ψ*¯*i* → *G*, where *ψ* and *ψ*¯ are the standard model particles on the brane (vector, spinor, scalar) and *G* is a state of mass *mn* from the graviton Kaluza-Klein tower. Production of Kaluza-Klein gravitons is calculated

where *Tμν* is the energy-momentum of particles on the brane. Evolution of energy density of matter on the brane is determined from the Boltzmann equation (Gorbunov & Rubakov, 2008;

> *d*3*k*2 (2*π*)32*E*2

> > *κ*2 8 *h*2 *n*(0)*s*

where *Ai* = *As*, *Av*, *Af* = 2/3, 4, 1 for scalars, vectors and fermions and *s*2 = (*k*1 + *k*2)2. The sum over the graviton states is transformed to the integral as in (62). Integrating over the

annihilation reaction. The square of the annihilation amplitude is (Langlois & Sorbo, 2003)

−*g h*¯ *μν*(0, *x*)*Tμν*(*x*),

*f i* 1(*E*1)*f i*

2(*E*2)|*Mi*

*n*|

2(2*π*)4*δ*4(*k*1 + *k*2 − *p*).

*n* is the amplitude of the

4, (76)

(75)

 2 *π*

argument of the Bessel function

asymptotics

giving

Solution of the equation

$$h\_m^{\prime\prime}(y) - 4\mu^2 h\_m(y) + e^{2\mu|y|} m\_<^2 h\_m(y) = 0\tag{65}$$

is

$$h\_m^{<}(y) = \mathbb{C}\_1 \mathbf{l}\_2(\tilde{m}e^{\mu|y|}) + \mathbb{C}\_2 \mathbf{N}\_2(\tilde{m}e^{\mu|y|}),\tag{66}$$

where

$$
\tilde{m} = \frac{m\_{<}}{\mu}.
$$

The terms with *δ*(*y*) are taken into account by the boundary condition

$$\left[\frac{d\hbar\_m(y)}{dy} + \frac{2}{\vec{m}}\hbar\_m(y)\right]\_{y=0\_+} = 0$$

which yields the relation *C*1 *J*1(*m*˜) + *C*2*N*1(*m*˜) = 0. The eigenfunctions (66) take the form

$$h\_{m\_{\prec}}^{<}(y) = \mathbb{C}\left[ (\mathcal{N}\_1(\mathfrak{m})I\_2(\mathfrak{m}e^{\mu|y|}) - I\_1(\mathfrak{m})\mathcal{N}\_2(\mathfrak{m}e^{\mu|y|})) \right].\tag{67}$$

Introducing the functions

$$f\_k(y) = \mathcal{N}\_1(\tilde{m}) f\_k(\tilde{m}e^{\mu|y|}) - f\_1(\tilde{m}) \mathcal{N}\_k(\tilde{m}e^{\mu|y|})$$

we obtain the norm of *hm*

$$||h\_{m,\cdot}^{<}||^2 = 2\mathbb{C}^2 \int\_0^{\overline{g}} e^{2\mu y} f\_2^2(y) dy = \frac{\mathbb{C}^2}{\mu} \left[ \varepsilon^{2\mu g} \left( f\_2^2(\bar{y}) - f\_1(\bar{y}) f\_3(\bar{y}) \right) - \left( f\_2^2(0) - f\_1(0) f\_3(0) \right) \right]. \tag{68}$$

Typical masses (energies) of the emitted Kaluza-Klein gravitons are of order of temperature of the Universe *T*. In the case of small *μ* ∼ 10−12 GeV we have *m*/*μ* � 1. Substituting the asymptotics of the Bessel functions, we obtain

$$||h\_{m\_{\prec}}^{<}||^{2} = \frac{\mathsf{C}^{2}}{\mu} \left(\frac{2}{\pi \mathfrak{m}}\right)^{2} (e^{\mu \mathcal{g}} - 1). \tag{69}$$

In the region *y* > *y*¯ we use the metric in (59) with the increasing exponent. The equations for the eigenmodes *h*> is

$$h\_m''(y) - 4\mu^2 h\_m(y) + e^{-2\mu|y|} m\_>^2 h\_m(y) = 0. \tag{70}$$

The eigenfunctions are

$$\mathcal{H}\_{m\_{\geq}}^{>}(y) = \tilde{\mathcal{C}}\_1 \mathcal{I}\_2(\mathfrak{m}e^{-\mu y}) - \tilde{\mathcal{C}}\_2 \mathcal{N}\_2(\mathfrak{m}e^{-\mu y}),\tag{71}$$

where *m*˜ = *m*>/*μ*. For large *y*, such that *me*˜ −*μy* � 1, the function *N*2(*me*˜ −*μy*) ∼ (*me*˜ −*μy*)−2 rapidly increases, and to have normalizable eigenfunctions we set *C*˜ 2 = 0. The norm of the function *h*> *m*(*y*) is

$$\langle ||h\_{m\_>}^{>}||^2 = \tilde{\mathcal{C}}\_1^2 \int\_{\mathcal{G}} e^{-2\mu y} f\_2^2(\tilde{m}e^{-\mu y}) dy = \tilde{\mathcal{C}}\_1^2 \frac{e^{-2\mu g}}{2\mu} [f\_2^2(\tilde{m}e^{-\mu g}) - f\_1(\tilde{m}e^{-\mu g})] \text{J} \tag{72}$$

At temperatures *T* � *μ* at which production of gravitons is sufficiently intensive, the argument of the Bessel function

$$\tilde{m}e^{-\mu g} \sim \frac{T}{\mu} \left(\frac{\rho}{\mu}\right)^{1/2} \sim \frac{T}{\mu}g\_\*^{1/2}(T)\frac{T^2}{\mu M\_{pl}}.$$

is large and we can substitute the asymptotics of the Bessel functions

$$h\_{m\_{\geq}}^{\geq}(y) = \tilde{\mathfrak{C}}\_{1} \sqrt{\frac{2}{\pi \tilde{m}}} e^{\mu \mathcal{G}/2} \cos(\tilde{m}(e^{\mu \mathcal{G}} - 1)). \tag{73}$$

Instead of sewing the oscillating functions *h*< *m*(*y*) and *h*> *m*(*y*), we sew the envelopes of their asymptotics

$$\frac{2\mathcal{C}}{\pi \pi \tilde{m}\_{<}} e^{-\mu \mathcal{G}/2} = \tilde{\mathcal{C}}\_1 \left(\frac{2}{\pi \tilde{m}\_{>}}\right)^{1/2} e^{\mu \mathcal{G}/2} \tag{74}$$

giving

12 Will-be-set-by-IN-TECH

*m*˜ = *m*< *μ* .

> 2 *m*˜ *hm*(*y*)

which yields the relation *C*1 *J*1(*m*˜) + *C*2*N*1(*m*˜) = 0. The eigenfunctions (66) take the form

(*N*1(*m*˜)*J*2(*me*˜ *μ*|*y*|

2*μ*|*y*| *m*2

) + *C*2*N*2(*me*˜ *μ*|*y*|

*y*=0+

= 0

) − *J*1(*m*˜)*N*2(*me*˜ *μ*|*y*|

) − *J*1(*m*˜)*Nk*(*me*˜ *μ*|*y*|

2 (*y*¯) − *f*1(*y*¯)*f*3(*y*¯)

Typical masses (energies) of the emitted Kaluza-Klein gravitons are of order of temperature of the Universe *T*. In the case of small *μ* ∼ 10−12 GeV we have *m*/*μ* � 1. Substituting the

> 2 *πm*˜

In the region *y* > *y*¯ we use the metric in (59) with the increasing exponent. The equations for

1 *J*2(*me*˜ −*μy*) − *C*˜

where *m*˜ = *m*>/*μ*. For large *y*, such that *me*˜ −*μy* � 1, the function *N*2(*me*˜ −*μy*) ∼ (*me*˜ −*μy*)−2

1 2 *e*−2*μy*¯ 2*μ* [*J* 2 2 (*me*˜ −*μy*¯

2 (*e*

−2*μ*|*y*| *m*2 )

)

 − *f* 2

<*hm*(*y*) = 0 (65)

), (66)

. (67)

2 (0) − *f*1(0)*f*3(0)

*μy*¯ − 1). (69)

>*hm*(*y*) = 0. (70)

2*N*2(*me*˜ −*μy*), (71)

) − *J*1(*me*˜ −*μy*¯

2 = 0. The norm of the

)]. (72)

)*J*3(*me*˜ −*μy*¯

 .

(68)

Solution of the equation

is

where

*h*��

*h*<

*m*(*y*) − 4*μ*2*hm*(*y*) + *e*

*m*(*y*) = *C*1 *J*2(*me*˜ *μ*|*y*|

The terms with *δ*(*y*) are taken into account by the boundary condition

 *dhm*(*y*) *dy* +

2 (*y*)*dy* = *C*2

*h*��

*h*>

*m*> (*y*) = *C*˜

rapidly increases, and to have normalizable eigenfunctions we set *C*˜

2 (*me*˜ −*μy*)*dy* = *C*˜

*fk*(*y*) = *N*1(*m*˜)*Jk*(*me*˜ *μ*|*y*|

*μ e* 2*μy*¯ *f* 2

*m*< ||2 = *C*2

*m*(*y*) − 4*μ*2*hm*(*y*) + *e*

*μ*

*h*<

Introducing the functions

we obtain the norm of *hm*

*y*¯

0 *e* 2*μy f* 2

asymptotics of the Bessel functions, we obtain

*m*< ||2 = 2*C*2

the eigenmodes *h*> is

The eigenfunctions are

*m*(*y*) is

1 ∞

*y*¯ *e* −2*μy J* 2

*m*> ||2 = *C*˜2

function *h*>

*m*< (*y*) = *C*

$$
\tilde{C}\_1 = C e^{-\mu \bar{y}} \sqrt{\frac{2}{\pi}} \frac{\sqrt{\tilde{m}\_>}}{\tilde{m}\_<}
$$

The norm (72) is smaller than (69). Effectively, we neglect the contribution from the region *y* > *y*¯ and impose the condition *hm*(*y*¯) = 0, which yields us the spectrum (60). Using the norm (69), we obtain the normalized eigenmode *hm*(0) (61).

#### **8. Production of Kaluza-Klein gravitons**

In this section we calculate the rate of production of Kaluza-Klein gravitons in interactions of particles of the hot matter on the brane in the radiation-dominated period. The leading contribution to this process is given by the annihilation reactions *ψi* + *ψ*¯*i* → *G*, where *ψ* and *ψ*¯ are the standard model particles on the brane (vector, spinor, scalar) and *G* is a state of mass *mn* from the graviton Kaluza-Klein tower. Production of Kaluza-Klein gravitons is calculated with the interaction Lagrangian

$$I = \kappa \int d^4x \sqrt{-\overline{\mathfrak{g}}} \, h\_{\mu\nu}(0, \mathfrak{x}) T^{\mu\nu}(\mathfrak{x}) .$$

where *Tμν* is the energy-momentum of particles on the brane. Evolution of energy density of matter on the brane is determined from the Boltzmann equation (Gorbunov & Rubakov, 2008; Kolb & Turner, 1990)

$$\frac{d\hat{\rho}}{dt} + 4H\hat{\rho} = -\sum\_{\text{n}} \sum\_{\text{i}} \int \frac{d^3 p}{(2\pi)^3} \int \frac{d^3 k\_1}{(2\pi)^3 2\mathcal{E}\_1} \frac{d^3 k\_2}{(2\pi)^3 2\mathcal{E}\_2} f\_1^i(\mathcal{E}\_1) f\_2^i(\mathcal{E}\_2) |M\_{\text{n}}^i|^2 (2\pi)^4 \delta^4(k\_1 + k\_2 - p). \tag{75}$$

Here *f i* are the Bose/Fermi distributions of colliding particles and *Mi n* is the amplitude of the annihilation reaction. The square of the annihilation amplitude is (Langlois & Sorbo, 2003)

$$|M\_n^i|^2 = A\_i \frac{\kappa^2}{8} h\_n^2(0) s^4,\tag{76}$$

where *Ai* = *As*, *Av*, *Af* = 2/3, 4, 1 for scalars, vectors and fermions and *s*2 = (*k*1 + *k*2)2. The sum over the graviton states is transformed to the integral as in (62). Integrating over the

**9. Graviton emission in the bulk in the period of late cosmology and**

To estimate the effect of the graviton emission on nucleosynthesis we solve perturbatively the system of the generalized Friedmann equation (51) and the 5D conservation equation (52). In the period of late cosmology the leading approximation of the Friedmann equation coincides with that in the standard cosmological model. Supposing that the terms with the bulk energy-momentum tensor do not alter the leading-order results significantly, we treat these terms perturbatively. The Weyl radiation term and the term quadratic in radiation

Graviton Emission in the Bulk and Nucleosynthesis in a Model with Extra Dimension 357

*H*2 � 2*μρ* − *I*04 − *I*44. (84)

))*a*−4(0, *t*

(*μtl*)3 − 1

�

(*μt*)3

48 1

2

2

,

(*μtl*)2 − 1

) = (85)

(*μt*)2

*ρ*2(*t*) (87)

*ρ*2(*t*) (88)

2 and 1/*μtl* � (8*ρ*(*tl*)/*μ*)1/2

3 (91)

, (86)

*ij*, we obtain

�

) = *μρ*(*t*)*A*44

 180 *g*∗*π*2

 180 *g*∗*π*2

*I*04 � −2*μρ*(*t*) · 0.0048 (89) *I*44 � 2*μρ*(*t*) · 0.00094. (90)

)(*μ* + *ρ*(*t*

�

**nucleosynthesis**

and

*I*44 = 2*κ*2

Here we substituted

3*a*4(0, *t*)

 *t tl dt*� *T*ˇ 44(*t* � )(2*μρ*(*t* �

*κ*2*T*ˇ

energy density can be treated perturbatively also.

Using the expressions (83) for the components *T*ˇ

*I*04 = 2*κ*2

*μρ*(*t*)*A*04 1

3*a*4(0, *t*)

 *t tl dt*� *T*ˇ 04(0, *t* �

12 1 *μtl* − 1 *μt* + 1 288 1

04 = *A*04*ρ*2 = −315 *A ζ*(9/2)*ζ*(7/2)

*κ*2*T*44 = *A*44*ρ*2 = 3 *A ζ*(9/2)*ζ*(7/2)

*A*04 � −0.126. In the leading approximation *ρ*(*t*) � 1/8*μt*

In the radiation-dominated period,the 5D conservation equation is

(5.1 ÷ 1.6) · 102*GeV*. For *ρ*(*tl*)/*μ* ∼ 0.1 we obtain

))1/2*a*−4(0, *t*

29*π*3

4*π*4

The integrals have a strong dependence on the value of the lower limit. Therefore, in the integrand the slowly varying functions can be taken at the times when all the Standard model degrees of freedom are relativistic. In this period *A* = 166.2, *g*∗(*T*) = 106.7 and

Taking *ρ*(*tl*)/*μ* ∼ 0.1 ÷ 0.001 and *μ* ∼ 10−12*GeV*, we have 1/*μtl* � 0.9 ÷ 0.09 and *Tl* ∼

*ρ*˙ + 4*Hρ* = − *A*04*ρ*2

*H*¯ 2 = 2*μρ*¯.

Let *ρ*¯ and *H*¯ be the energy density and the Hubble function in the leading order

The Friedmann equation can be written as

angular variables of the momenta of interacting particles, we obtain Boltzmann equation in the form

$$\frac{d\hat{\rho}}{dt} + 4H\dot{\rho} = -\frac{\kappa^2 \sum\_i A\_i}{8(2\pi)^4} \int \frac{dme^{\mu g}}{\mu} h\_m^2(0) m^4 \int dk\_1 dk\_2 f(k\_1) f(k\_2) (k\_1 + k\_2) \theta\left(1 - \left|1 - \frac{m^2}{2k\_1 k\_2}\right|\right). \tag{77}$$

Integrating over *m*, we obtain (cf. (Hebecker & March-Russel, 2001; Langlois & Sorbo, 2003))

$$\begin{split} \frac{d\hat{\rho}}{dt} + 4H\hat{\rho} &= -\frac{\kappa^2 \sum\_i A\_i}{8(2\pi)^4} \int\_0^{(2k\_1k\_2)^{1/2}} dm m^4 \int dk\_1 dk\_2 f(k\_1) f(k\_2) (k\_1 + k\_2) \\ &= -\frac{\kappa^2 A 315 \zeta(9/2) \zeta(7/2) T^8}{\pi^3 2^8}, \end{split} \tag{78}$$

where the sum extends over relativistic degrees of freedom

$$A = \sum\_{i} A\_{i} = \frac{2g\_{\text{s}}}{3} + g\_{f}(1 - 2^{-7/2})(1 - 2^{-9/2}) + 4g\_{\text{v}}.\tag{79}$$

In the high-energy period, when all the standard model degrees of freedom are relativistic, *A* = 166, 2.

Eq.(78) has the same form as Eq.(52) and the right hand side of (78) can be identified with the component *T*ˇ 04 of the bulk energy-momentum tensor (Langlois & Sorbo, 2003).

The energy-momentum tensor of the emitted gravitons is taken in the form of free radiation of massless particles (Langlois & Sorbo, 2003)

$$\check{T}^{ij}(\mathbf{x}) = \int d^5p \sqrt{-g} \delta(p\_i p^i) \tilde{f}(\mathbf{x}, p) p^i p^j \,. \tag{80}$$

where ˜ *f*(*x*, *p*) is the phase space density of the distribution function of gravitons.

Expanding the graviton momentum *pi* in the orthonormal basis *vi b*, *ni b* (19), we have

$$p^i = E\upsilon^i\_b + mn^i\_b + p^a e^i\_{a\nu}$$

where *pi pi* = −*E*2 + *m*2 + *pa pa*. Substituting these expressions in (80), we obtain

$$\check{T}\_{nu} = u^{i}n^{j}T\_{i\bar{j}} = -\frac{1}{2} \int d^{3}pdm \, mf(E, m, p^{a})\Big|\_{E=\sqrt{m^{2}+p^{2}}}\tag{81}$$

The interaction tern in (75) can be rewritten as

$$\frac{1}{(2\pi)^5} \int d^3p \frac{dm}{\pi} \int \frac{d^3k\_1}{2\mathcal{E}\_1} \frac{d^3k\_2}{2\mathcal{E}\_2} f\_1^i(\mathcal{E}\_1) f\_2^i(\mathcal{E}\_2) \sum\_i |M\_n^i|^2 \delta^4(k\_1 + k\_2 - p) \tag{82}$$

Comparing (81) and (82), one determines *f*(*E*, *m*, *pa*) and calculates the other components of *T*ˇ *ij*

$$\check{T}\_{\rm{III}}(0,t) = \frac{3A\zeta(9/2)\zeta(7/2)}{2\pi^4} \kappa^2 T^8, \qquad \check{T}\_{\rm{III}}(0,t) = \frac{21A\zeta(9/2)\zeta(7/2)}{8\pi^4} \kappa^2 T^8. \tag{83}$$

In the basis (19), the components *T*ˇ *nu* and *T*ˇ *nn* are equal to *T*ˇ 04 and *T*ˇ 44 .

## **9. Graviton emission in the bulk in the period of late cosmology and nucleosynthesis**

To estimate the effect of the graviton emission on nucleosynthesis we solve perturbatively the system of the generalized Friedmann equation (51) and the 5D conservation equation (52). In the period of late cosmology the leading approximation of the Friedmann equation coincides with that in the standard cosmological model. Supposing that the terms with the bulk energy-momentum tensor do not alter the leading-order results significantly, we treat these terms perturbatively. The Weyl radiation term and the term quadratic in radiation energy density can be treated perturbatively also.

The Friedmann equation can be written as

$$
\Delta H^2 \simeq 2\mu\rho - I\_{04} - I\_{44}.\tag{84}
$$

Using the expressions (83) for the components *T*ˇ *ij*, we obtain

$$I\_{04} = \frac{2\mathbf{x}^2}{3a^4(0, t)} \int\_{t\_l}^t dt' \check{T}\_{04}(0, t') (\mu + \rho(t')) a^{-4}(0, t') = \tag{85}$$

$$\mu \rho(t) A\_{04} \left( \frac{1}{12} \left( \frac{1}{\mu t\_l} - \frac{1}{\mu t} \right) + \frac{1}{288} \left( \frac{1}{(\mu t\_l)^3} - \frac{1}{(\mu t)^3} \right) \right),$$

and

14 Will-be-set-by-IN-TECH

angular variables of the momenta of interacting particles, we obtain Boltzmann equation in

Integrating over *m*, we obtain (cf. (Hebecker & March-Russel, 2001; Langlois & Sorbo, 2003))

In the high-energy period, when all the standard model degrees of freedom are relativistic,

Eq.(78) has the same form as Eq.(52) and the right hand side of (78) can be identified with the

04 of the bulk energy-momentum tensor (Langlois & Sorbo, 2003). The energy-momentum tensor of the emitted gravitons is taken in the form of free radiation

−*gδ*(*pi pi*

*f*(*x*, *p*) is the phase space density of the distribution function of gravitons.

*b* + *mni*

Comparing (81) and (82), one determines *f*(*E*, *m*, *pa*) and calculates the other components of

) ˜ *f*(*x*, *p*)*pi*

*b* + *pae i a*,

*d*3 *pdm mf*(*E*, *m*, *pa*)

2(*E*2)∑ *i* |*Mi n*|

*nn* are equal to *T*ˇ

*dmm*4

*dk*1*dk*2 *f*(*k*1)*f*(*k*2)(*k*1 + *k*2)*θ*

3 + *gf*(1 − 2−7/2)(1 − 2−9/2) + 4*gv*. (79)

*pj*

 *E*=

*uu*(0, *t*) = 21 *Aζ*(9/2)*ζ*(7/2)

04 and *T*ˇ

44 .

*b*, *ni*

√*m*2+*p*2

, (80)

(81)

*b* (19), we have

2*δ*4(*k*1 + *k*2 − *p*) (82)

8*π*4 *κ*2*T*8. (83)

 1 −

*dk*1*dk*2 *f*(*k*1)*f*(*k*2)(*k*1 + *k*2) (78)

1 − *m*2 2*k*1*k*2  . (77)

(2*k*1*k*2)1/2

0

the form

*A* = 166, 2.

component *T*ˇ

where ˜

*T*ˇ *ij*

*dt* + 4*Hρ*ˆ = −*κ*2 ∑*i Ai*

*dρ*ˆ

8(2*π*)4

 *dmeμy*¯ *μ*

*dt* + 4*Hρ*ˆ = −*κ*2 ∑*i Ai*

*A* = ∑ *i*

of massless particles (Langlois & Sorbo, 2003)

*T*ˇ *nu* = *ui nj*

 *d*3 *p dm π*

1 (2*π*)5

In the basis (19), the components *T*ˇ

*T*ˇ

The interaction tern in (75) can be rewritten as

*nn*(0, *t*) = 3*Aζ*(9/2)*ζ*(7/2)

= −*κ*2*A*315*ζ*(9/2)*ζ*(7/2)*T*8

where the sum extends over relativistic degrees of freedom

*h*2 *m*(0)*m*4

8(2*π*)4

*Ai* = 2*gs*

*T*ˇ*ij*(*x*) =

Expanding the graviton momentum *pi* in the orthonormal basis *vi*

*Tij* = −1 2

 *d*3*k*1 2*E*1

 *d*5 *p*

*pi* = *Evi*

where *pi pi* = −*E*2 + *m*2 + *pa pa*. Substituting these expressions in (80), we obtain

*d*3*k*2 2*E*2 *f i* 1(*E*1)*f i*

2*π*4 *κ*2*T*8, *T*ˇ

*nu* and *T*ˇ

*π*328 ,

*dρ*ˆ

$$I\_{44} = \frac{2\kappa^2}{3a^4(0,t)} \int\_{t\_l}^t dt' \breve{T}\_{44}(t') (2\mu\rho(t'))^{1/2} a^{-4}(0,t') = \frac{\mu\rho(t)A\_{44}}{48} \left(\frac{1}{(\mu t\_l)^2} - \frac{1}{(\mu t)^2}\right), \tag{86}$$

Here we substituted

$$\kappa^2 \dddot{\Upsilon}\_{04} = A\_{04} \rho^2 = -\frac{315 \, A \, \zeta(9/2) \zeta(7/2)}{2^9 \pi^3} \left(\frac{180}{g\_\* \pi^2}\right)^2 \rho^2(t) \tag{87}$$

$$
\kappa^2 T\_{44} = A\_{44} \rho^2 = \frac{3A \zeta(9/2)\zeta(7/2)}{4\pi^4} \left(\frac{180}{g\_\*\pi^2}\right)^2 \rho^2(t) \tag{88}
$$

The integrals have a strong dependence on the value of the lower limit. Therefore, in the integrand the slowly varying functions can be taken at the times when all the Standard model degrees of freedom are relativistic. In this period *A* = 166.2, *g*∗(*T*) = 106.7 and *A*04 � −0.126. In the leading approximation *ρ*(*t*) � 1/8*μt* 2 and 1/*μtl* � (8*ρ*(*tl*)/*μ*)1/2 Taking *ρ*(*tl*)/*μ* ∼ 0.1 ÷ 0.001 and *μ* ∼ 10−12*GeV*, we have 1/*μtl* � 0.9 ÷ 0.09 and *Tl* ∼ (5.1 ÷ 1.6) · 102*GeV*. For *ρ*(*tl*)/*μ* ∼ 0.1 we obtain

$$I\_{04} \simeq -2\mu\rho(t) \cdot 0.0048 \tag{89}$$

$$I\_{44} \simeq 2\mu\rho(t) \cdot 0.00094.\tag{90}$$

In the radiation-dominated period,the 5D conservation equation is

$$
\phi + 4H\rho = -\frac{A\_{04}\rho^2}{3} \tag{91}
$$

Let *ρ*¯ and *H*¯ be the energy density and the Hubble function in the leading order

$$
\bar{H}^2 = 2\mu\bar{\rho}.
$$

where

the freezing temperature

*ρ*¯(*tn*)

 *n p n*

*δX*4 � 2

integral in the Boltzmann equation and the expressions for *T*ˇ

brane (Hebecker & March-Russel, 2001; Langlois & Sorbo, 2003)

These gravitons do not contribute to the component *T*ˇ

equations assuming that this cancellation takes place.

reflected, but contribute to the component *T*ˇ

 *t tc dt*� *T*ˇ 04(*t* � )(*ρ*(*t* �

is the term *T*ˇ(*b*)

*T*ˇ(*b*)

− 2*κ*2 3*a*4(0, *t*)

cosmology remain qualitatively valid in the early cosmological period.

which is also a very small number (Iofa, 2011).

*μ* = 4*π*3*g*∗(*Tn*) 90

is very small. The equilibrium value of the *n* − *p* ratio at the freezing temperature is

= exp

(*n*/*p* + 1)2 ln *n*

Substituting *Tn* ∼ 10−3*GeV* and *μM*3 ∼ (*μMpl*)2 ∼ 1014*GeV*4, we find that the ratio *δTn*/*Tn*

Graviton Emission in the Bulk and Nucleosynthesis in a Model with Extra Dimension 359

Substituting *δ*(*n*/*p*)*n* = (*n*/*p*)ln(*n*/*p*)*nδTn*/*Tn*, we obtain variation of *X*4 under variation of

**10. Estimate of the graviton emission in the bulk in the period of early cosmology** In the period of early cosmology in the Friedmann equation the *ρ*2 term is dominant, i.e. *ρ*(*t*)/*μ* > 1. For the value of *μ* ∼ 10−12*GeV* assumed in the present study the characteristic temperatures of the period of early cosmology are above 5 · 102*GeV*. Not much is known about physics at such temperatures. To make a crude estimate of the effect of the graviton emission on the nucleosynthesis we adopt a conservative point of view assuming that the collision

The new phenomenon in the early cosmological period is that some of the emitted gravitons can return to the brane and be again reflected in the bulk with a different momentum.

) + *μ*) + *T*ˇ

Here the initial time *tc* is the time of reheating. Numerical estimates and considerations from the Vaidya model (Langlois; Sorbo & Rodriguez-Martinez, 2002; Langlois & Sorbo, 2003) suggest that at the period of early cosmology the dominant contributions from *T*04, *T*44 and

44 mutually cancel. Below we solve the system of the Friedmann and 5D conservation

Let *ρ*¯ and *H*¯ be the energy density of matter on the brane and the Hubble function calculated in the model without the graviton emission in the period of early cosmology *H*¯ 2 � *ρ*¯2(*t*). For

*ρ* = *ρ*¯ + *ρ*2, *H* = *H*¯ + *H*2,

the equation of state of the hot plasma *ρ*¯ = *p*¯/3, we have *ρ*¯(*t*) � 1/4*t*. Defining

44 representing the energy-momentum tensor of gravitons bouncing back to the

*H*2(*t*) = *ρ*2(*t*) + 2*μρ*(*t*) + *μρw*(*t*) (100)

44(*t* � )*H*(*t* � ) − *T*ˇ(*b*) 44 (*t* � )*H*(*t* � ) *a*4(0, *t* � ).

*T*4 *n*

> .

> > *δTn Tn*

−(*mn* − *mp*) *Tn*

*p*

 *n p n*

*μM*3 . (98)

*ij* calculated in the period of late

04, because they are not produced, but

44. The new ingredient in the Friedmann equation

(99)

Defining

$$
\rho = \bar{\rho} + \rho\_{1\nu} \qquad \qquad H = \bar{H} + H\_{1\nu}
$$

and separating in (84) and (91) the leading-order terms, we obtain

$$2\bar{H}H\_1 = 2\mu\rho\_1 - \frac{2\bar{\rho}}{3} \int dt' \left[A\_{04}(\mu + \bar{\rho}) + A\_{44}\bar{H}\right] \bar{\rho}(t')\tag{92}$$

$$
\rho\_1 + 4\bar{H}\rho\_1 + 4\bar{\rho}H\_1 = \frac{A\_{04}\bar{\rho}^2}{3}. \tag{93}
$$

Substituting in the above system *ρ*¯(*t*) � 1/8*μt* 2 and performing integration, we have

$$H\_1(t) \simeq \frac{\mu}{H} \rho\_1 - \frac{\mathcal{H}A\_{04}}{48\mu} \left[ \frac{1}{t\_l} - \frac{1}{t} + \frac{1}{24\mu^2} \left( \frac{1}{t\_l^3} - \frac{1}{t^3} \right) \right] - \frac{\mathcal{H}A\_{44}}{192\mu^2} \left( \frac{1}{t\_l^2} - \frac{1}{t^2} \right) . \tag{94}$$

Substituting in (92) expression (94) for *H*1 and noting that *H*¯ = 1/2*t*, we obtain

$$
\rho\_1 + \frac{3}{t} \rho\_1 = \frac{A\_{04}}{192 \mu^2 t^3} \left[ \frac{1}{t\_l} + \frac{1}{24 \mu^2} \left( \frac{1}{t\_l^3} - \frac{1}{t^3} \right) \right] + \frac{A\_{44}}{768 \mu^3 t^3} \left( \frac{1}{t\_l^2} - \frac{1}{t^2} \right).
$$

Solving this equation, we find

$$\rho\_1(t) = \frac{C\_1}{t^3} + \frac{A\_{04}}{192\mu^2 t^3} \left[ \frac{t - t\_l}{t\_l} + \frac{1}{24\mu^2} \left( \frac{t - t\_l}{t\_l^3} - \frac{1}{2} \left( \frac{1}{t\_l^2} - \frac{1}{t^2} \right) \right) \right] + \frac{A\_{44}}{768\mu^3 t^3} \left( \frac{t}{t\_l^2} - \frac{2}{t\_l} + \frac{1}{t} \right) \tag{95}$$

The constant *C*1 can be determined by sewing solutions of Friedmann equation in the periods of early and late cosmologies. For a moment we set *C*1 = 0, i.e. we look for a contribution from the period of late cosmology. For *H*1 we obtain

$$H\_1(t) = -\frac{A\_{04}}{1536\mu^3 t^2} \left(\frac{1}{t\_l^2} - \frac{1}{t^2}\right) + \frac{A\_{44}}{192\mu^2 t^2} \left(-\frac{1}{t\_l} + \frac{1}{t}\right) \simeq -\bar{\rho}(t) \left(\frac{A\_{04}}{192\mu^2 t\_l^2} + \frac{A\_{44}}{24\mu t\_l}\right). \tag{96}$$

The expressions (95) and (96) show that corrections to the leading terms are small, i.e. the perturbative approach is justified. Note that the leading term proportional to *A*04/*μtl* was canceled in *H*1.

The mass fraction of 4*He* produced in primordial nucleosynthesis is (Gorbunov & Rubakov, 2008; Kolb & Turner, 1990)

$$\mathcal{X}\_4 = \frac{2(n/p)}{(n/p) + 1} \prime$$

where the ratio *n*/*p* is taken at the end of nucleosunthesis. Characteristic temperature at the onset of the period of nucleosynthesis (freezing temperature *Tn* of the reaction *n* ↔ *p*), estimated as the temperature at which the reaction rate ∼ *GFT*5 is approximately equal to the Hubble parameter *GFT*5 *n* ∼ *H* (Gorbunov & Rubakov, 2008; Kolb & Turner, 1990), is *Tn* ∼ 10−3*GeV*. The difference of the freezing temperatures in the models with and without the account of the graviton emission is

$$\frac{\delta T\_n}{T\_n} \simeq \frac{H\_1}{5\bar{H}} \simeq \frac{1}{5} \sqrt{\frac{\bar{\rho}(t\_n)}{2\mu}} \left( \frac{A\_{04}}{192\mu^2 t\_l^2} + \frac{A\_{44}}{24\mu t\_l} \right),\tag{97}$$

where

16 Will-be-set-by-IN-TECH

*ρ* = *ρ*¯ + *ρ*1, *H* = *H*¯ + *H*1,

1 24*μ*2

> 1 *t* 3 *l*

 1 *t* 3 *l*

> − 1 *t*3

> > 1 *t* 2 *l*

− 1 2

The constant *C*1 can be determined by sewing solutions of Friedmann equation in the periods of early and late cosmologies. For a moment we set *C*1 = 0, i.e. we look for a contribution

> − 1 *tl* + 1 *t*

The expressions (95) and (96) show that corrections to the leading terms are small, i.e. the perturbative approach is justified. Note that the leading term proportional to *A*04/*μtl* was

The mass fraction of 4*He* produced in primordial nucleosynthesis is (Gorbunov & Rubakov,

*X*4 = 2(*n*/*p*) (*n*/*p*) + 1

where the ratio *n*/*p* is taken at the end of nucleosunthesis. Characteristic temperature at the onset of the period of nucleosynthesis (freezing temperature *Tn* of the reaction *n* ↔ *p*), estimated as the temperature at which the reaction rate ∼ *GFT*5 is approximately equal to the

10−3*GeV*. The difference of the freezing temperatures in the models with and without the

*A*04 192*μ*2*t* 2 *l* + *A*44 24*μtl*

 *ρ*¯(*tn*) 2*μ*

,

*n* ∼ *H* (Gorbunov & Rubakov, 2008; Kolb & Turner, 1990), is *Tn* ∼

*dt*� [*A*04(*μ* + *ρ*¯) + *A*44*H*¯ ] *ρ*¯(*t*

− 1 *t*3

> +

> > − 1 *t*2

�

 1 *t* 2 *l*

− 1 *t*2

− 1 *t*2 .

> *t t* 2 *l* − 2 *tl* + 1 *t*

*A*44 768*μ*3*t*3

*A*04 192*μ*2*t* 2 *l* + *A*44 24*μtl*

3 . (93)

 1 *t* 2 *l*

2 and performing integration, we have

− *HA*¯ 44 192*μ*2

*A*44 768*μ*3*t*3

 +

� −*ρ*¯(*t*)

, (97)

) (92)

. (94)

(95)

. (96)

and separating in (84) and (91) the leading-order terms, we obtain

*ρ*˙1 + 4*H*¯ *ρ*1 + 4*ρ*¯*H*1 = *A*04*ρ*¯2

 1 *tl* − 1 *t* +

 1 *tl* + 1 24*μ*2

3

Substituting in (92) expression (94) for *H*1 and noting that *H*¯ = 1/2*t*, we obtain

 *t* − *tl t* 3 *l*

*A*44 192*μ*2*t*2

2*HH*¯ 1 = 2*μρ*1 − 2*ρ*¯

Substituting in the above system *ρ*¯(*t*) � 1/8*μt*

*H*¯ *ρ*1 − *HA*¯ 04 48*μ*

*ρ*1 = *A*04 192*μ*2*t*3

> *t* − *tl tl* + 1 24*μ*2

from the period of late cosmology. For *H*1 we obtain

− 1 *t*2 +

 1 *t* 2 *l*

*H*1(*t*) � *μ*

*ρ*˙1 + 3 *t*

Solving this equation, we find

*A*04 192*μ*2*t*3

1536*μ*3*t*2

*t*3 +

*H*1(*t*) = − *A*04

2008; Kolb & Turner, 1990)

Hubble parameter *GFT*5

account of the graviton emission is

*δTn Tn*

� *H*1 5*H*¯ � 1 5

canceled in *H*1.

*ρ*1(*t*) = *C*1

Defining

$$\frac{\bar{\rho}(t\_n)}{\mu} = \frac{4\pi^3 g\_\*(T\_n)}{90} \frac{T\_n^4}{\mu M^3}. \tag{98}$$

Substituting *Tn* ∼ 10−3*GeV* and *μM*3 ∼ (*μMpl*)2 ∼ 1014*GeV*4, we find that the ratio *δTn*/*Tn* is very small. The equilibrium value of the *n* − *p* ratio at the freezing temperature is

$$\left(\frac{n}{p}\right)\_n = \exp\left[-\frac{(m\_n - m\_p)}{T\_n}\right].$$

Substituting *δ*(*n*/*p*)*n* = (*n*/*p*)ln(*n*/*p*)*nδTn*/*Tn*, we obtain variation of *X*4 under variation of the freezing temperature

$$
\delta X\_4 \simeq \frac{2}{(n/p+1)^2} \ln \left(\frac{n}{p}\right) \left(\frac{n}{p}\right)\_n \frac{\delta T\_n}{T\_n} \tag{99}
$$

which is also a very small number (Iofa, 2011).

#### **10. Estimate of the graviton emission in the bulk in the period of early cosmology**

In the period of early cosmology in the Friedmann equation the *ρ*2 term is dominant, i.e. *ρ*(*t*)/*μ* > 1. For the value of *μ* ∼ 10−12*GeV* assumed in the present study the characteristic temperatures of the period of early cosmology are above 5 · 102*GeV*. Not much is known about physics at such temperatures. To make a crude estimate of the effect of the graviton emission on the nucleosynthesis we adopt a conservative point of view assuming that the collision integral in the Boltzmann equation and the expressions for *T*ˇ *ij* calculated in the period of late cosmology remain qualitatively valid in the early cosmological period.

The new phenomenon in the early cosmological period is that some of the emitted gravitons can return to the brane and be again reflected in the bulk with a different momentum. These gravitons do not contribute to the component *T*ˇ 04, because they are not produced, but reflected, but contribute to the component *T*ˇ 44. The new ingredient in the Friedmann equation is the term *T*ˇ(*b*) 44 representing the energy-momentum tensor of gravitons bouncing back to the brane (Hebecker & March-Russel, 2001; Langlois & Sorbo, 2003)

$$\begin{split}H^{2}(t)&=\rho^{2}(t)+2\mu\rho(t)+\mu\rho\_{w}(t)\\&-\frac{2\kappa^{2}}{3a^{4}(0,t)}\int\_{t\_{\zeta}}^{t}dt'\left[\check{T}\_{04}(t')(\rho(t')+\mu)+\check{T}\_{44}(t')H(t')-\check{T}\_{44}^{(b)}(t')H(t')\right]a^{4}(0,t').\end{split}\tag{100}$$

Here the initial time *tc* is the time of reheating. Numerical estimates and considerations from the Vaidya model (Langlois; Sorbo & Rodriguez-Martinez, 2002; Langlois & Sorbo, 2003) suggest that at the period of early cosmology the dominant contributions from *T*04, *T*44 and *T*ˇ(*b*) 44 mutually cancel. Below we solve the system of the Friedmann and 5D conservation equations assuming that this cancellation takes place.

Let *ρ*¯ and *H*¯ be the energy density of matter on the brane and the Hubble function calculated in the model without the graviton emission in the period of early cosmology *H*¯ 2 � *ρ*¯2(*t*). For the equation of state of the hot plasma *ρ*¯ = *p*¯/3, we have *ρ*¯(*t*) � 1/4*t*. Defining

$$
\rho = \overline{\rho} + \rho\_{2\prime} \qquad \qquad H = \bar{H} + H\_{2\prime}
$$

In the period of late cosmology the term *C*1/*t*

ln 1 *μtc* − 1 2 − *A*44 48 +

Δ*H*1 = *μC*1/*Ht* ¯ 3

Δ*H*1 = 16*ρ*¯

Here *ρ*¯(*t*) � 1/8*μt*

From (94) we would have

At time of the nucleosynthesis

contribution

 *μ A*04 24 3 generates in Eq.(94) for *H*1 the contribution

*b*1*μ* 24 ln 1 *μtc* . (110)

. (111)

. (112)

− *b*2 24*tc* + *b*2*μ* 12 +

*A*04 + *A*44 96*tc*

(*μt*)3 � 3 · 10−4

(*μtc*)(*μt*)

*T*4 *n* (*μMpl*)2 .

2 + *b*1 ln 1

2 = *ρ*2 + 2*μρ* the equation for the brane trajectory can be

*μ*2 + *H*2 , (113)

. (114)

*H*

*μtc*

(*μt*)3*tc* .

2 is the matter energy density in the period of late cosmology.

1

*H*¯ 2 Δ*ρ*1 � 1.2 · 10−3

If the term (*A*04 + *A*44)/96*tc* ∼ 3 · 10−4/*tc* was not canceled, it would produce in *ρ*1 the

Graviton Emission in the Bulk and Nucleosynthesis in a Model with Extra Dimension 361

Δ*ρ*1 = *A*04 + *A*44 96*tc*

> Δ*H*1 *H*¯ = *μ*

> > 1

 *A*04

too large a value, and would contradict the experimental data.

*R*˙ *b*/*Rb*

= *μ*2*R*2

*dRb dTb*

*ξ μ*2*R*2 *b* + *R*˙ 2 *b*

*bR*˙ *b*

*dRb dTb*

Assuming that the large terms in (110) cancel, we have

Δ*H*1 = 2*ρ*¯(*tn*) 3

**11. Gravitons bouncing to the brane**

Using Friedmann equation,

obtain

written as (Langlois & Sorbo, 2003)

8(*μtn*)2 � 4*π*3*g*∗(*Tn*)

90

For *Tn* ∼ 10−3*GeV*, we have *μtn* ∼ 1012. From (111) we obtain Δ*H*1(*tn*)/*H*¯ (*tn*) � 4, which is

Because in the period of nucleosynthesis Δ*H*1/*H*¯ ∼ (*ρ*¯(*tn*)/*μ*)1/2 is a small number,

In this section, in the framework of the brane moving in the static space-time we present arguments that in the period of early cosmology gravitons can bounce back to the brane.

where *β* = *sign*(*Rb*(*t*)). Expanding the right hand side of (113) in powers of *μ*/*ρ* < 1, we

� *ξβμ*2*R*2 *b* 1 − *μ*2 2*ρ*2

= *ξβμ*2*R*2 *b*

ln 1 *μtc* − 1 2 + *A*04

contribution from the early cosmology would result in a small variation of *δX*4/*X*4.

we obtain

$$2\bar{H}H\_{2} \simeq 2\bar{\rho}\rho\_{2} - \frac{2\bar{\rho}}{3} \int\_{t\_{\zeta}}^{t} dt' \left[ \left( A\_{04}(\mu + \bar{\rho}) + A\_{44}\bar{H} \right) \bar{\rho}(t') - \kappa^{2} \check{T}\_{44}^{(b)}(t') \right] \tag{101}$$

$$
\phi\_2 + 4H\rho\_2 + 4\bar{\rho}H\_2 = \frac{A\_{04}\bar{\rho}^2}{3}.\tag{102}
$$

The equation for *ρ*2 is

$$\rho\_2 + \frac{2}{t}\rho\_2 - \frac{1}{t} \left[ \frac{A\_{04}}{12} \left( \mu \ln \frac{t}{t\_c} + \frac{1}{4} \left( \frac{1}{t\_c} - \frac{1}{t} \right) \right) + \frac{A\_{44}}{48} \left( \frac{1}{t\_c} - \frac{1}{t} \right) \right] + \frac{1}{12t} \int\_{t\_c}^{t} \mathbf{x}^2 \tilde{T}\_{44}^{(b)}(t')dt' = \frac{A\_{04}}{48t^2} \text{.(103)}$$

Integrating (102) with the initial condition *ρ*2(*tc*) = 0, we obtain

$$\begin{split} \rho\_{2}(t) &= \frac{A\_{04}\mu}{24} \left( \ln \frac{t}{t\_{\mathcal{L}}} - \frac{1}{2} + \frac{t\_{\mathcal{c}}^{2}}{2t^{2}} \right) + \frac{A\_{04} + A\_{44}}{96t\_{\mathcal{C}}} \left( 1 - \frac{t\_{\mathcal{C}}^{2}}{t^{2}} \right) - \frac{A\_{44}}{48} \left( \frac{1}{t} - \frac{t\_{\mathcal{C}}}{t^{2}} \right) \\ &- \frac{1}{12t^{2}} \int\_{t\_{\mathcal{L}}}^{t} dy \int\_{t\_{\mathcal{L}}}^{y} dx \mathbf{x}^{2} \dot{\Upsilon}\_{44}^{(b)}(\mathbf{x}) \end{split} \tag{104}$$

The time *tc* of reheating is estimated for the reheating temperature *TR* ∼ 5 · 106*GeV* (Mielczarek, 2009). Using the relation (53) and substituting *ρ*¯(*tc*)/*μ* � 1/4*μtc*, we have

$$\frac{1}{4\mu t\_c} \simeq \frac{4\pi^3 g\_\*(T\_R) T\_R^4}{90\mu M^3} \sim 9 \cdot 10^{14} \text{ J}$$

It follows that *tl*/*tc* ∼ 1/*μtc* � 3.5 · 1015.

In (104) there is a large term (*A*04 + *A*44)/*tc*. Omitting the small terms, we have

$$\rho\_2(t) \simeq \frac{A\_{04}\mu}{24} \left( \ln \frac{t}{t\_c} - \frac{1}{2} \right) + \frac{A\_{04} + A\_{44}}{96t\_c} - \frac{A\_{44}}{48} \frac{1}{t} - \frac{1}{12t^2} \int\_{t\_c}^{t} dy y \int\_{t\_c}^{y} dx \boldsymbol{\kappa}^2 \dot{T}\_{44}^{(b)}(\mathbf{x}) \tag{105}$$

On dimensional grounds at small *t* the term *κ*2*T*ˇ(*b*) 44 (*t*) has the following structure

$$\kappa^2 \dot{T}\_{44}^{(b)}(t) = \frac{b\_2}{t^2} + \frac{b\_1 \mu}{t} + b\_0 \mu^2 + \cdots \tag{106}$$

Performing integration of the last term in (105) and taking *t* ∼ *tl*, we obtain

$$\rho\_2(t\_l) \simeq \frac{A\_{04}\mu}{24} \left( \ln \frac{t\_l}{t\_\complement} - \frac{1}{2} \right) - \frac{A\_{44}}{48t\_l} + \frac{A\_{04} + A\_{44}}{96t\_\complement} - \frac{b\_2}{24t\_\complement} + \frac{b\_2}{12t\_l} + \frac{b\_1\mu}{24} \ln \frac{t\_l}{t\_\complement}.\tag{107}$$

Next, we equate *ρ*2(*tl*) and *C*1/*t* 3 *l* in (95). At the times *t* ∼ *tl*, where *μtl* ∼ 1, we have

$$\mathcal{C}\_{1}\mu^{3} \simeq \mu \left(\frac{A\_{04}}{24}\left(\ln\frac{1}{\mu t\_{\mathcal{C}}} - \frac{1}{2}\right) - \frac{A\_{44}}{48}\right) + \frac{A\_{04} + A\_{44}}{96t\_{\mathcal{C}}} - \frac{b\_{2}}{24t\_{\mathcal{C}}} + \frac{b\_{2}\mu}{12} + \frac{b\_{1}\mu}{24}\ln\frac{1}{\mu t\_{\mathcal{C}}}.\tag{108}$$

The term *C*1/*t* 3 is

$$\frac{C\_1}{t^3} \sim \frac{1}{\mu^2 t^3} \left[ \mu \left( \frac{A\_{04}}{24} \left( \ln \frac{1}{\mu t\_c} - \frac{1}{2} \right) - \frac{A\_{44}}{48} \right) + \frac{A\_{04} + A\_{44}}{96 t\_c} - \frac{b\_2}{24 t\_c} + \frac{b\_2 \mu}{12} + \frac{b\_1 \mu}{24} \ln \frac{1}{\mu t\_c} \right]. \tag{109}$$

In the period of late cosmology the term *C*1/*t* 3 generates in Eq.(94) for *H*1 the contribution Δ*H*1 = *μC*1/*Ht* ¯ 3

$$\Delta H\_1 = 16\bar{\rho} \left[ \mu \left( \frac{A\_{04}}{24} \left( \ln \frac{1}{\mu t\_c} - \frac{1}{2} \right) - \frac{A\_{44}}{48} \right) + \frac{A\_{04} + A\_{44}}{96 t\_c} - \frac{b\_2}{24 t\_c} + \frac{b\_2 \mu}{12} + \frac{b\_1 \mu}{24} \ln \frac{1}{\mu t\_c} \right]. \tag{110}$$

Here *ρ*¯(*t*) � 1/8*μt* 2 is the matter energy density in the period of late cosmology.

If the term (*A*04 + *A*44)/96*tc* ∼ 3 · 10−4/*tc* was not canceled, it would produce in *ρ*1 the contribution

$$
\Delta \rho\_1 = \frac{A\_{04} + A\_{44}}{96t\_c} \frac{1}{(\mu t)^3} \simeq \frac{3 \cdot 10^{-4}}{(\mu t)^3 t\_c}.
$$

From (94) we would have

18 Will-be-set-by-IN-TECH

(*A*04(*μ* + *ρ*¯) + *A*44*H*¯ ) *ρ*¯(*t*

 1 *tc* − 1 *t* + 1 12*t*

*A*04 + *A*44 96*tc*

*R* 90*μM*3 ∼ 9 · 1014.

The time *tc* of reheating is estimated for the reheating temperature *TR* ∼ 5 · 106*GeV* (Mielczarek, 2009). Using the relation (53) and substituting *ρ*¯(*tc*)/*μ* � 1/4*μtc*, we have

> − *A*44 48 1 *t* − 1 12*t*2

> > *b*1*μ*

*A*04 + *A*44 96*tc*

> *A*04 + *A*44 96*tc*

> > *A*04 + *A*44 96*tc*

− *b*2 24*tc* + *b*2 12*tl* + *b*1*μ* 24 ln *tl tc*

*l* in (95). At the times *t* ∼ *tl*, where *μtl* ∼ 1, we have

− *b*2 24*tc* + *b*2*μ* 12 +

− *b*2 24*tc* + *b*2*μ* 12 +

� 4*π*3*g*∗(*TR*)*T*4

In (104) there is a large term (*A*04 + *A*44)/*tc*. Omitting the small terms, we have

*A*04 + *A*44 96*tc*

44 (*t*) = *b*2

Performing integration of the last term in (105) and taking *t* ∼ *tl*, we obtain

*t*2 +

 1 − *t* 2 *c t*2 − *A*44 48

> *t tc dyy y tc*

44 (*t*) has the following structure

*t* + *b*0*μ*2 + ··· (106)

�

3 . (102)

 *t tc*

) − *κ*2*T*ˇ(*b*) 44 (*t* � )

> *κ*2*T*ˇ(*b*) 44 (*t* � )*dt*� = *A*04 48*t*2 ,(103)

> > 1 *t* − *tc t*2

*dxκ*2*T*ˇ(*b*)

*b*1*μ* 24 ln 1 *μtc*

*b*1*μ* 24 ln 1 *μtc*

44 (*x*) (105)

. (107)

. (108)

. (109)

(101)

(104)

we obtain

*ρ*˙2+ 2 *t ρ*2−1 *t*

The equation for *ρ*2 is

 *A*04 12

*ρ*2(*t*) = *A*04*μ* 24

> *t tc dyy y tc*

It follows that *tl*/*tc* ∼ 1/*μtc* � 3.5 · 1015.

 ln *t tc* − 1 2 +

On dimensional grounds at small *t* the term *κ*2*T*ˇ(*b*)

− 1 12*t*2

*ρ*2(*t*) � *A*04*μ* 24

*ρ*2(*tl*) � *A*04*μ*

*C*1*μ*3 � *μ*

The term *C*1/*t*

*C*1 *t*3 � 1 *μ*2*t*3 24

Next, we equate *ρ*2(*tl*) and *C*1/*t*

 *A*04 24

3 is

 *μ A*04 24

 ln *tl tc* − 1 2 − *A*44 48*tl* +

 ln 1 *μtc* − 1 2 − *A*44 48 +

> ln 1 *μtc* − 1 2 − *A*44 48 +

 *μ* ln *t tc* + 1 4 1 *tc* −1 *t* + *A*44 48

> ln *t tc* − 1 2 + *t* 2 *c* 2*t*2 +

2*HH*¯ 2 � 2*ρρ*¯ 2 − 2*ρ*¯

3 *t tc dt*�

Integrating (102) with the initial condition *ρ*2(*tc*) = 0, we obtain

*dxκ*2*T*ˇ(*b*) 44 (*x*)

> 1 4*μtc*

*κ*2*T*ˇ(*b*)

3

*ρ*˙2 + 4*H*¯ *ρ*2 + 4*ρ*¯*H*2 = *A*04*ρ*¯2

$$\frac{\Delta H\_1}{\bar{H}} = \frac{\mu}{\bar{H}^2} \Delta \rho\_1 \simeq \frac{1.2 \cdot 10^{-3}}{(\mu t\_c)(\mu t)}.\tag{111}$$

At time of the nucleosynthesis

$$\frac{1}{8(\mu t\_n)^2} \simeq \frac{4\pi^3 g\_\*(T\_n)}{90} \frac{T\_n^4}{(\mu M\_{pl})^2}.$$

For *Tn* ∼ 10−3*GeV*, we have *μtn* ∼ 1012. From (111) we obtain Δ*H*1(*tn*)/*H*¯ (*tn*) � 4, which is too large a value, and would contradict the experimental data.

Assuming that the large terms in (110) cancel, we have

$$
\Delta H\_1 = \frac{2\bar{\rho}(t\_\text{\(t\)})}{3} \left[ A\_{04} \left( \ln \frac{1}{\mu t\_\text{c}} - \frac{1}{2} \right) + \frac{A\_{04}}{2} + b\_1 \ln \frac{1}{\mu t\_\text{c}} \right]. \tag{112}
$$

Because in the period of nucleosynthesis Δ*H*1/*H*¯ ∼ (*ρ*¯(*tn*)/*μ*)1/2 is a small number, contribution from the early cosmology would result in a small variation of *δX*4/*X*4.

#### **11. Gravitons bouncing to the brane**

In this section, in the framework of the brane moving in the static space-time we present arguments that in the period of early cosmology gravitons can bounce back to the brane.

Using Friedmann equation, *R*˙ *b*/*Rb* 2 = *ρ*2 + 2*μρ* the equation for the brane trajectory can be written as (Langlois & Sorbo, 2003)

$$\frac{d\mathcal{R}\_b}{dT\_b} = \frac{\mu^2 \mathcal{R}\_b^2 \dot{\mathcal{R}}\_b}{\xi \sqrt{\mu^2 \mathcal{R}\_b^2 + \dot{\mathcal{R}}\_b^2}} = \xi \beta \mu^2 \mathcal{R}\_b^2 \frac{H}{\sqrt{\mu^2 + H^2}},\tag{113}$$

where *β* = *sign*(*Rb*(*t*)). Expanding the right hand side of (113) in powers of *μ*/*ρ* < 1, we obtain

$$\frac{dR\_b}{dT\_b} \simeq \xi \beta \mu^2 R\_b^2 \left(1 - \frac{\mu^2}{2\rho^2}\right). \tag{114}$$

**12. Conclusion**

the brane to the bulk.

leading-order approximation are small.

calculated in the standard cosmological model.

two pictures.

is small.

In a model of 3-brane embedded in 5D space-time we calculated graviton emission of interacting hot matter on the brane in the bulk. Reliable calculations can be performed in the period of late cosmology, when *ρ*(*T*)/*μ* < 1, where *ρ*(*T*) is the normalized radiation energy density of matter on the brane, *μ* = √−Λ/6 is the scale of the warping factor in the metric, Λ is the 5D cosmological constant. For *μ* ∼ 10−12*GeV*, which we adopted in this paper, the limiting temperatures of the Universe at which the approximation of late cosmology is valid are of order *Tl* ∼ 5 · 102*GeV*. In the period of late cosmology it was possible to make a number of approximations, which enabled us to obtain the analytic expression for the energy loss from

Graviton Emission in the Bulk and Nucleosynthesis in a Model with Extra Dimension 363

The 5D model of the present paper can be treated in two alternative approaches - with moving brane in the static 5D bulk and with static brane in 5D bulk with time-dependent metric. Each picture proves to be useful for particular problems. We establish explicit connection between

From the system of the Einstein equations containing the terms due to graviton emission we obtained the generalized Friedmann equation. The Einstein equations and Friedmann equation were solved perturbatively, the leading-order solution being that without the graviton emission. We have shown that in the period of late cosmology corrections to the

Graviton emission changes cosmological evolution of matter on the brane and thus production of light elements in primordial nucleosynthesis. Solving the system of the generalized Friedmann and the 5D energy conservation equations, which included the graviton emission terms, we found the difference of the mass fractions of 4*He* produced in primordial

nucleosynthesis calculated in the models with and without the graviton emission

*δX*4 *X*4

∼ *C ρ*(*Tn*) *μ* .

Here *ρ*(*T*) is is the radiation energy density of matter on the brane, *Tn* ∼ 10−3*GeV* is the freezing temperature of the reaction *p* ↔ *n*, and *C* � 1 is a small number. The ratio *ρ*(*Tn*)/*μ* ∼ 10−26 is a small number, and thus correction to abundance of 4*He* due to the graviton emission

To estimate the effect of the graviton emission in the period of early cosmology on Helium production, we assumed that the collision integral in the Boltzmann equation obtained for the period of late cosmology qualitatively retains its form in the period of early cosmology. As the upper limit of temperatures of matter on the brane at which we estimate the graviton production, we take the reheating temperature ∼ 106*GeV*. It appears that straightforward calculation of abundance of 4*He* produced in the early cosmology is too large to comply with the data. In the period of early cosmology a new effect becomes important (Hebecker & March-Russel, 2001; Langlois & Sorbo, 2003): gravitons emitted from the brane can bounce back to the brane. Assuming that the large terms in the components of the energy-momentum tensors from the emitted and bouncing gravitons mutually cancel, we find that the graviton emission in the period of early cosmology yields a small correction to abundance of 4*He*

Using that *ρ*(*t*) = *ρ*(*t*1)(*Rb*(*t*1)/*Rb*(*t*))4, we have

$$\left(1 - \frac{\mu^2}{2\rho^2}\right)^{-1} \simeq 1 + \frac{\mu^2 R\_b^8(t)}{2\rho^2(t\_1) R\_b^8(t\_1)}.$$

Integrating Eq. (113) with the initial conditions *Rb* = *Rb*(*t*1), *Tb* = *Tb*(*t*1), we obtain

$$\xi \beta \mu^2 (T\_b(t) - T\_b(t\_1)) = -\frac{1}{R\_b(t)} + \frac{1}{R\_b(t\_1)} + \frac{\mu^2}{14(\rho(t\_1)R\_b^4(t\_1))^2} \left(R\_b^7(t) - R\_b^7(t\_1)\right). \tag{115}$$

Gravitons propagate along the null geodesics which are found from the geodesic equations (14)-(16). The tangent vectors to a null geodesic satisfy the relation

$$\int\_{\mathbb{S}} g\_{ij} \frac{d\mathbf{x}^i}{d\lambda} \frac{d\mathbf{x}^j}{d\lambda} = \mathbf{0},\tag{116}$$

where, to distinguish the case of the null geodesics, we relabeled the affine parameter from *y* to *λ*. Substituting in (116) solution (17), we obtain that *CR* = 0. The tangent vectors to a null geodesic are

$$\frac{d\tilde{T}}{d\lambda} = \frac{\mathbb{C}^{T}}{f(\tilde{\mathbb{R}})}, \qquad \frac{d\tilde{\mathbb{X}}^{a}}{d\lambda} = \frac{\mathbb{C}^{a}}{\mu^{2}\tilde{\mathbb{R}}^{2}}, \qquad \frac{d\tilde{\mathbb{R}}}{d\lambda} = \nu|\mathbb{C}^{T}|\left(1-\gamma\right)^{1/2}.\tag{117}$$

where *tilde* indicates that the point is at the null geodesic, *ν* = ±1. From the equations (117) we obtain

$$\frac{d\tilde{R}}{d\tilde{T}} = \nu \kappa (1 - \gamma)^{1/2} \mu^2 \tilde{R}^2 \tag{118}$$

where *κ* ≡ *sign*(*CT*). Integrating this equation with the initial conditions *R*˜ = *Rb*(*t*1), *T*˜ = *Tb*(*t*1), we have

$$\frac{1}{R\_b(t\_1)} - \frac{1}{\tilde{R}} = \nu \mathfrak{x} (1 - \gamma)^{1/2} \mu^2 (\tilde{T} - T\_b(t\_1)). \tag{119}$$

To find, if at a time *t* the graviton trajectory returns to the brane world sheet, i.e. *R*˜ = *Rb*(*t*) and *T*˜ = *Tb*(*t*), we combine equations (115) and (119). In the case *ξβ* = *νκ* we obtain

$$\left[\left(1-\gamma\right)^{-1/2}-1\right]\left(\frac{1}{R\_b(t\_1)}-\frac{1}{R\_b(t)}\right) = \frac{\mu^2}{14\rho^2(t\_1)}\frac{1}{R\_b(t\_1)}\left(\left(\frac{R\_b(t)}{R\_b(t\_1)}\right)^7-1\right). \tag{120}$$

Eq. (120) means that graviton emitted from the brane at the time *t*1 has bounced back on the brane at the time *t*. Setting *z* ≡ *Rb*(*t*1)/*Rb*(*t*), and *γ*ˆ/2 ≡ [(1 − *γ*)−1/2 − 1], and noting that in the period of early cosmology *μ*/*ρ*(*t*) � 4*μt* � 1, we rewrite (120) as

$$\frac{(1-z^7)}{z^7(1-z)} = \frac{14\hat{\gamma}\rho^2(t\_1)}{\mu^2} \simeq \frac{14\hat{\gamma}}{(4\mu t\_1)^2} \tag{121}$$

The function (1 − *z*7)*z*−7(1 − *z*)−1 is monotone decreasing with the minimum at *z* = 1 , and Eq. (121) has a unique solution provided 2*γ*ˆ/(4*μt*1)2 > 1.

From these relations, using that in the period of early cosmology *z* = *Rb*(*t*1)/*Rb*(*t*) � (*t*1/*t*)1/4, is calculated the time of the bounce. The time of the bounce is small, if *z* ∼ 1, or *γ*/(4*μt*)2 ∼ 1, that is if gravitons are emitted at small angles to the brane.

## **12. Conclusion**

20 Will-be-set-by-IN-TECH

� 1 + *μ*2*R*8

*Rb*(*t*1) + *μ*2

Gravitons propagate along the null geodesics which are found from the geodesic equations

*dxj*

where, to distinguish the case of the null geodesics, we relabeled the affine parameter from *y* to *λ*. Substituting in (116) solution (17), we obtain that *CR* = 0. The tangent vectors to a null

where *tilde* indicates that the point is at the null geodesic, *ν* = ±1. From the equations (117)

where *κ* ≡ *sign*(*CT*). Integrating this equation with the initial conditions *R*˜ = *Rb*(*t*1), *T*˜ =

To find, if at a time *t* the graviton trajectory returns to the brane world sheet, i.e. *R*˜ = *Rb*(*t*)

= *μ*2 14*ρ*2(*t*1)

Eq. (120) means that graviton emitted from the brane at the time *t*1 has bounced back on the brane at the time *t*. Setting *z* ≡ *Rb*(*t*1)/*Rb*(*t*), and *γ*ˆ/2 ≡ [(1 − *γ*)−1/2 − 1], and noting that in

The function (1 − *z*7)*z*−7(1 − *z*)−1 is monotone decreasing with the minimum at *z* = 1 , and

From these relations, using that in the period of early cosmology *z* = *Rb*(*t*1)/*Rb*(*t*) � (*t*1/*t*)1/4, is calculated the time of the bounce. The time of the bounce is small, if *z* ∼ 1,

and *T*˜ = *Tb*(*t*), we combine equations (115) and (119). In the case *ξβ* = *νκ* we obtain

*z*7(1 − *z*) = 14*γρ*ˆ 2(*t*1)

or *γ*/(4*μt*)2 ∼ 1, that is if gravitons are emitted at small angles to the brane.

*Rb*(*t*)

*μ*2*R*˜ 2 , *dR*˜

*b*(*t*)

*b*(*t*1) .

*b*(*t*1))2

*dλ* = *ν*|*CT*|(1 − *γ*)

*dT*˜ = *νκ*(1 − *γ*)1/2*μ*2*R*˜ 2, (118)

*R*˜ = *νκ*(1 − *γ*)1/2*μ*2(*T*˜ − *Tb*(*t*1)). (119)

 *Rb*(*t*) *Rb*(*t*1) 7 − 1

(4*μt*1)2 (121)

1 *Rb*(*t*1)

*μ*2 � 14*γ*ˆ

 *R*7

*dλ* = 0, (116)

*b*(*t*) − *R*7

*b*(*t*1)

1/2 , (117)

. (120)

. (115)

2*ρ*2(*t*1)*R*8

14(*ρ*(*t*1)*R*4

−1

Integrating Eq. (113) with the initial conditions *Rb* = *Rb*(*t*1), *Tb* = *Tb*(*t*1), we obtain

1

*gij dxi dλ*

*dλ* = *Ca*

Using that *ρ*(*t*) = *ρ*(*t*1)(*Rb*(*t*1)/*Rb*(*t*))4, we have

*ξβμ*2(*Tb*(*t*) − *Tb*(*t*1)) = − 1

*dT*˜ *dλ* = *CT f*(*R*˜)

geodesic are

we obtain

*Tb*(*t*1), we have

[(1 − *γ*)−1/2 − 1]

 1 − *μ*2 2*ρ*2

*Rb*(*t*) +

(14)-(16). The tangent vectors to a null geodesic satisfy the relation

, *dx*˜*a*

1 *Rb*(*t*1) − 1

1

*Rb*(*t*1) − 1

the period of early cosmology *μ*/*ρ*(*t*) � 4*μt* � 1, we rewrite (120) as

(1 − *z*7)

Eq. (121) has a unique solution provided 2*γ*ˆ/(4*μt*1)2 > 1.

*dR*˜

In a model of 3-brane embedded in 5D space-time we calculated graviton emission of interacting hot matter on the brane in the bulk. Reliable calculations can be performed in the period of late cosmology, when *ρ*(*T*)/*μ* < 1, where *ρ*(*T*) is the normalized radiation energy density of matter on the brane, *μ* = √−Λ/6 is the scale of the warping factor in the metric, Λ is the 5D cosmological constant. For *μ* ∼ 10−12*GeV*, which we adopted in this paper, the limiting temperatures of the Universe at which the approximation of late cosmology is valid are of order *Tl* ∼ 5 · 102*GeV*. In the period of late cosmology it was possible to make a number of approximations, which enabled us to obtain the analytic expression for the energy loss from the brane to the bulk.

The 5D model of the present paper can be treated in two alternative approaches - with moving brane in the static 5D bulk and with static brane in 5D bulk with time-dependent metric. Each picture proves to be useful for particular problems. We establish explicit connection between two pictures.

From the system of the Einstein equations containing the terms due to graviton emission we obtained the generalized Friedmann equation. The Einstein equations and Friedmann equation were solved perturbatively, the leading-order solution being that without the graviton emission. We have shown that in the period of late cosmology corrections to the leading-order approximation are small.

Graviton emission changes cosmological evolution of matter on the brane and thus production of light elements in primordial nucleosynthesis. Solving the system of the generalized Friedmann and the 5D energy conservation equations, which included the graviton emission terms, we found the difference of the mass fractions of 4*He* produced in primordial nucleosynthesis calculated in the models with and without the graviton emission

$$\frac{\delta X\_4}{X\_4} \sim C \sqrt{\frac{\rho(T\_n)}{\mu}}.$$

Here *ρ*(*T*) is is the radiation energy density of matter on the brane, *Tn* ∼ 10−3*GeV* is the freezing temperature of the reaction *p* ↔ *n*, and *C* � 1 is a small number. The ratio *ρ*(*Tn*)/*μ* ∼ 10−26 is a small number, and thus correction to abundance of 4*He* due to the graviton emission is small.

To estimate the effect of the graviton emission in the period of early cosmology on Helium production, we assumed that the collision integral in the Boltzmann equation obtained for the period of late cosmology qualitatively retains its form in the period of early cosmology. As the upper limit of temperatures of matter on the brane at which we estimate the graviton production, we take the reheating temperature ∼ 106*GeV*. It appears that straightforward calculation of abundance of 4*He* produced in the early cosmology is too large to comply with the data. In the period of early cosmology a new effect becomes important (Hebecker & March-Russel, 2001; Langlois & Sorbo, 2003): gravitons emitted from the brane can bounce back to the brane. Assuming that the large terms in the components of the energy-momentum tensors from the emitted and bouncing gravitons mutually cancel, we find that the graviton emission in the period of early cosmology yields a small correction to abundance of 4*He* calculated in the standard cosmological model.

**1. Introduction**

deflection foreseen by GR with an accuracy of ∼ 10%.

in the future, by other projects presently under study.

First experimental evidences of General Relativity (GR) came from Astrometry. Its measurements not only had put to evidence the existence of the effect of an excess of perihelion precession in the motion of the Solar System objects which couldn't be explained in terms of the Newtonian Mechanics (Le Verrier, 1859) but also provided a striking confirmation of Einstein's theory through the experiment conducted at Sobral and Principe by Dyson, Eddington & Davidson (1920) during the Solar Eclipse of 1919, which measured the light

**17**

*Italy*

**Putting Einstein to Test –**

Alberto Vecchiato

**Astrometric Experiments from Space,**

**Fundamental Physics and Local Cosmology**

*INAF (National Institute for Astrophysics), Astronomical Observatory of Torino*

GR, however, has never been the only theory claiming to provide a detailed description of the Gravitational interaction beyond the Newtonian limit. Although GR is still the most favoured alternative, either because of the experimental results or of its simplicity, several other theories have been proposed. Examples of such theories are those where the field equations contain not only the metric tensor of GR, but also a scalar field coupled with the metric itself (Will, 2006), or the so-called fourth-order theories of gravity, where the scalar of curvature *R* in the

Tests of gravity theories within the Solar System are usually analysed in the framework of the so-called *Parametrized Post-Newtonian framework* which enables the comparison of several theories through the estimation of the value of a limited number of parameters. Among these parameters, *γ* and *β* are of particular importance for astrometry since they are connected with the classical astrometric phenomena of the light deflection and of the excess of perihelion precession in the orbits of massive objects. Besides its immediate implication for the fundamental physics problem of characterizing the best gravity theory, a precise estimation of these parameters has important consequences on the interpetation of observational evidences at different scales in space and time up to cosmological scales. This is the reason why these Solar System astrometric experiments can also be intended as a kind of "local cosmology" tests which will be addressed from space by planned missions like Gaia (Perryman et al., 2001) and,

In the next section we will birefly recall the basic concepts standing at the foundations of the current gravity theories, and we will try to give a general overview of the "what" and

field equations is replaced by a more complex function of this quantity, *f*(*R*).

#### **13. References**

## **Putting Einstein to Test – Astrometric Experiments from Space, Fundamental Physics and Local Cosmology**

Alberto Vecchiato

*INAF (National Institute for Astrophysics), Astronomical Observatory of Torino Italy*

## **1. Introduction**

22 Will-be-set-by-IN-TECH

364 Astrophysics

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Durrer, R. (2005). Braneworlds, *XI Brazilian School of Cosmology and Gravitation, AIP Conf. Proc.*

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Hebecker, A. & March-Russel, J. (2001) Randall-Sundrum II Cosmology, AdS/CFT, and the

Iofa, M. (2009)a. Constraints on parameters of models with extra dimension from primordial

Iofa, M. (2009)b. Cosmological constraints on parameters of one-brane models with extra

Iofa, M. (2011). Graviton emission from the brane in the bulk in a model with extra dimension

Kolb, E. & Turner, M. (1990). *The Early Universe*. Addison-Wesley, Frontiers in Physics,

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Langlois, D.; Sorbo, L. & Rodriguez-Martinez, M. (2002). Cosmology of a brane radiating gravitons into the extra dimension. *Phys. Rev. Lett.* 89, 171301. [hep-th/0206146] Langlois, D. & Sorbo, L. (2003) Bulk gravitons from a cosmological brane. *Phys. Rev.* D68,

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Mukohyama, S., Shiromizu,T. & Maeda, K-i (2000). Global structure of exact cosmological solutions in the brane world. *Phys. Rev.* D62 (2000) 024028. [arXiv:hep-th/9912287] Randall, L. & Sundrum, R. (1999) An Alternative to Compactification. *Phys. Rev. Lett.* B83,

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**13. References**

First experimental evidences of General Relativity (GR) came from Astrometry. Its measurements not only had put to evidence the existence of the effect of an excess of perihelion precession in the motion of the Solar System objects which couldn't be explained in terms of the Newtonian Mechanics (Le Verrier, 1859) but also provided a striking confirmation of Einstein's theory through the experiment conducted at Sobral and Principe by Dyson, Eddington & Davidson (1920) during the Solar Eclipse of 1919, which measured the light deflection foreseen by GR with an accuracy of ∼ 10%.

GR, however, has never been the only theory claiming to provide a detailed description of the Gravitational interaction beyond the Newtonian limit. Although GR is still the most favoured alternative, either because of the experimental results or of its simplicity, several other theories have been proposed. Examples of such theories are those where the field equations contain not only the metric tensor of GR, but also a scalar field coupled with the metric itself (Will, 2006), or the so-called fourth-order theories of gravity, where the scalar of curvature *R* in the field equations is replaced by a more complex function of this quantity, *f*(*R*).

Tests of gravity theories within the Solar System are usually analysed in the framework of the so-called *Parametrized Post-Newtonian framework* which enables the comparison of several theories through the estimation of the value of a limited number of parameters. Among these parameters, *γ* and *β* are of particular importance for astrometry since they are connected with the classical astrometric phenomena of the light deflection and of the excess of perihelion precession in the orbits of massive objects. Besides its immediate implication for the fundamental physics problem of characterizing the best gravity theory, a precise estimation of these parameters has important consequences on the interpetation of observational evidences at different scales in space and time up to cosmological scales. This is the reason why these Solar System astrometric experiments can also be intended as a kind of "local cosmology" tests which will be addressed from space by planned missions like Gaia (Perryman et al., 2001) and, in the future, by other projects presently under study.

In the next section we will birefly recall the basic concepts standing at the foundations of the current gravity theories, and we will try to give a general overview of the "what" and

2. the quantities *Rαβ* and *R* basically depend on the derivatives of the metric, so that the Poisson equation can be derived from Eq. (3) under the same weak-field and slow-motion

367 Putting Einstein to Test –

Another example of relativistic field theory of gravity is the Brans-Dicke theory. (Brans & Dicke, 1961) This theory is the simplest example of the class of the so-called *scalar-tensor* theories, in which the field equation is characterized by the presence of a scalar field *φ* coupled

The quantity *ω* is a coupling constant2 which gives a measure of the deviations of this theory from GR. From this equation, in fact, it can be shown that the relation between *φ* and the trace

so that Eq. (6) reduces to the GR field equation (3) when *ω* → ∞. The maini motivation at the basis of this theory was to have a gravity theory which could fully incorporate the Mach principle, according to which the inertial phenomena were caused by the relative motion of

The importance of scalar-tensor theories comes also from other more complex models introduced as a way to explain inflationary cosmological scenarios (like e.g. in Damour & Nordtvedt (1993)), and also in particle-physics-inspired models of unification such as string

We finally want to mention the so-called *f*(*R*) theories. This class of theories stem from the idea that the lagrangian L from which the field equations can be derived can be more general with respect to that of GR. The latter, in fact, assumes that it is simply equal to the Ricci scalar, i.e. L = *R*, while these theories relax this condition and take a generic dependence L = *f* (*R*). From a mathematical point of view, a consequence of this hypothesis is that the field equations are no more second-order differential equations, but fourth-order. From a physical point of view, the additional terms emerging into the field equation can be interpreted also as sources, rather than fields, and take the same role of *Tαβ*. From a phenomenological point of view, finally, these theories have been invoked to explain several observational phenomena like cosmological acceleration, galactic dynamics, gravitational lensing, etc. without the need to resort to non-barionic Dark Matter or to Dark Energy. A more detailed review of this kind of

As we have just pointed out in the previous section, several other theories alternative to GR has been formulated in order to describe how gravity works. This situation naturally calls for a framework in which it is possible to model the observations taking into account many viable theories at the same time. In this section we will briefly explain two of these frameworks: the

2 In more general versions of scalar-tensor theories *ω* is not a constant but a function of *φ*.

*φ* = 8*πG*

*c*4 *Tαβ* +

*ω φ*

*φ*,*αφ*,*β* − 1

(2*ω* + 3) *c*4 *T* (7)

2

*gαβφ*,*σφ*,*σ*

. (6)

+ *gαβφ* − *φ*,*α*;*β* = 8*πG*

Astrometric Experiments from Space, Fundamental Physics and Local Cosmology

assumptions.

*φ*  *Rαβ* − 1 2 *gαβR*

*T* of the stress-energy tensor is

with gravity in addition to the metric tensor

the mass in the Universe. (Ciufolini & Wheeler, 1995)

theories can be found in Capozziello & Faraoni (2011).

**3. The theoretical tools of the gravity arena**

theory (Damour, Piazza & Veneziano, 2002).

the "why", that is which are the alternatives available on the "theoretical market" and their motivations. In Section 3 we will show the "how", i.e. the theoretical methods which have been developed to make the comparison among the various gravity theories easier, while in the following one we will concentrate on the problem of how it is possible to model a generic astrometric observable in a relativistic consistent way. Section 5 will adress the experimental side of this problem giving examples of data reduction techniques and of some experiments planned or under study, and finally the next and last section will come to the conclusions.

### **2. Theoretical background**

The basic tenets of the so-called *calssical field theories* could be summarized, very shortly and roughly, as:

Mathematically, these two simple principles correspond to the *field equations* and to the *equations of motion* respectively.

The first and simplest theory of this kind is the newtonian theory of gravity, in which the Poisson equation

$$
\nabla^2 \mathcal{U} = 4\pi G\rho \tag{1}
$$

is the field equation where the gravitational field *U* originates from a mass density *ρ*, while the Newtonian law of dynamics

$$\frac{\text{d}^2 \text{x}^k}{\text{d}t^2} = -\frac{\partial \mathcal{U}}{\partial \text{x}^k} \tag{2}$$

are the equations of motion.

However, many other classical field theories aiminig at explaining how gravity works have been formulated. In GR the field equations write

$$R\_{a\beta} - \frac{1}{2}g\_{a\beta}R = \frac{8\pi G}{c^4}T\_{a\beta} \tag{3}$$

where the metric *gαβ* plays the role of the potential *U* and the stress-energy tensor *Tαβ* is the relativistic counterpart of the matter density *ρ*. 1 This statement can be better understood if one notices that:

1. under the assumption of weak gravitational fields and slow motion, the relativistic equations of motion approximate to

$$\frac{\text{d}^2 \text{x}^k}{\text{d}t^2} = -\frac{c^2}{2} \frac{\partial g\_{00}}{\partial \text{x}^k} \, \text{} \tag{4}$$

and the requirement that, within these hypothesis, they have to reproduce the newtonian ones implies that

$$g\_{00} \simeq -1 + \frac{2L}{c^2};\tag{5}$$

1 The definition of the stress-energy tensor is more general and takes into account that in GR any form of energy can contribute to the curvature of the space-time.

2 Will-be-set-by-IN-TECH

the "why", that is which are the alternatives available on the "theoretical market" and their motivations. In Section 3 we will show the "how", i.e. the theoretical methods which have been developed to make the comparison among the various gravity theories easier, while in the following one we will concentrate on the problem of how it is possible to model a generic astrometric observable in a relativistic consistent way. Section 5 will adress the experimental side of this problem giving examples of data reduction techniques and of some experiments planned or under study, and finally the next and last section will come to the conclusions.

The basic tenets of the so-called *calssical field theories* could be summarized, very shortly and

Mathematically, these two simple principles correspond to the *field equations* and to the

The first and simplest theory of this kind is the newtonian theory of gravity, in which the

is the field equation where the gravitational field *U* originates from a mass density *ρ*, while

d*t*2 = − *∂U*

However, many other classical field theories aiminig at explaining how gravity works have

where the metric *gαβ* plays the role of the potential *U* and the stress-energy tensor *Tαβ* is the

1. under the assumption of weak gravitational fields and slow motion, the relativistic

2 *∂g*00

2*U*

and the requirement that, within these hypothesis, they have to reproduce the newtonian

*g*00 � −1 +

1 The definition of the stress-energy tensor is more general and takes into account that in GR any form of

d2*xk* d*t*2 = −*c*2

*gαβR* = 8*πG*

d2*xk*

*Rαβ* − 1 2

∇2*U* = 4*πGρ* (1)

*∂xk* (2)

*c*4 *Tαβ* (3)

*∂xk* , (4)

*c*2 ; (5)

1 This statement can be better understood if

1. there exists something (the field source) producing a field;

2. this field tells to the particles how to move.

been formulated. In GR the field equations write

relativistic counterpart of the matter density *ρ*.

energy can contribute to the curvature of the space-time.

equations of motion approximate to

**2. Theoretical background**

*equations of motion* respectively.

the Newtonian law of dynamics

are the equations of motion.

roughly, as:

Poisson equation

one notices that:

ones implies that

2. the quantities *Rαβ* and *R* basically depend on the derivatives of the metric, so that the Poisson equation can be derived from Eq. (3) under the same weak-field and slow-motion assumptions.

Another example of relativistic field theory of gravity is the Brans-Dicke theory. (Brans & Dicke, 1961) This theory is the simplest example of the class of the so-called *scalar-tensor* theories, in which the field equation is characterized by the presence of a scalar field *φ* coupled with gravity in addition to the metric tensor

$$
\oint \left( R\_{a\beta} - \frac{1}{2} g\_{a\beta} R \right) + g\_{a\beta} \Box \phi - \phi\_{,a;\beta} = \frac{8\pi G}{c^4} T\_{a\beta} + \frac{\omega}{\phi} \left( \phi\_{,a} \phi\_{,\beta} - \frac{1}{2} g\_{a\beta} \phi\_{,\sigma} \phi^{,\sigma} \right) . \tag{6}$$

The quantity *ω* is a coupling constant2 which gives a measure of the deviations of this theory from GR. From this equation, in fact, it can be shown that the relation between *φ* and the trace *T* of the stress-energy tensor is

$$
\Box \phi = \frac{8\pi G}{\left(2\omega + 3\right)c^4} T \tag{7}
$$

so that Eq. (6) reduces to the GR field equation (3) when *ω* → ∞. The maini motivation at the basis of this theory was to have a gravity theory which could fully incorporate the Mach principle, according to which the inertial phenomena were caused by the relative motion of the mass in the Universe. (Ciufolini & Wheeler, 1995)

The importance of scalar-tensor theories comes also from other more complex models introduced as a way to explain inflationary cosmological scenarios (like e.g. in Damour & Nordtvedt (1993)), and also in particle-physics-inspired models of unification such as string theory (Damour, Piazza & Veneziano, 2002).

We finally want to mention the so-called *f*(*R*) theories. This class of theories stem from the idea that the lagrangian L from which the field equations can be derived can be more general with respect to that of GR. The latter, in fact, assumes that it is simply equal to the Ricci scalar, i.e. L = *R*, while these theories relax this condition and take a generic dependence L = *f* (*R*). From a mathematical point of view, a consequence of this hypothesis is that the field equations are no more second-order differential equations, but fourth-order. From a physical point of view, the additional terms emerging into the field equation can be interpreted also as sources, rather than fields, and take the same role of *Tαβ*. From a phenomenological point of view, finally, these theories have been invoked to explain several observational phenomena like cosmological acceleration, galactic dynamics, gravitational lensing, etc. without the need to resort to non-barionic Dark Matter or to Dark Energy. A more detailed review of this kind of theories can be found in Capozziello & Faraoni (2011).

## **3. The theoretical tools of the gravity arena**

As we have just pointed out in the previous section, several other theories alternative to GR has been formulated in order to describe how gravity works. This situation naturally calls for a framework in which it is possible to model the observations taking into account many viable theories at the same time. In this section we will briefly explain two of these frameworks: the

2 In more general versions of scalar-tensor theories *ω* is not a constant but a function of *φ*.

**Parameter Meaning Value in**

Astrometric Experiments from Space, Fundamental Physics and Local Cosmology

How much space-curvature is produced by unit rest mass?

How much nonlinearity in the superposition law for gravity?

Preferred-frame effects?

Violation of conservation of total momentum?

*γ*

*β*

*α*1

gravity.

GR Brans-Dicke Vector-Tensor *f* (*R*)

1 1 *β*� *β*�

0 0 *α*�

radar signals 2.3 · 10−5

2 + *ω γ*� *γ*�

1 0

2 0

1 1 + *ω*

*ξ* Preferred-location effects? 0 0 0 0

369 Putting Einstein to Test –

*α*3 0 0 0 0

*ζ*1 0 0 0 0 *ζ*2 0 0 0 0 *ζ*3 0 0 0 0 *ζ*4 0 0 0 0

Table 1. List of the PPN parameters, and of their meanings and values in some theories of

HIPPARCOS Light deflection Global Astrometry 3 · 10−3 VLBI Light deflection Radio Interferometry 4.5 · 10−4

Cassini Shapiro time delay Round-trip travel time of

Table 2. Current best estimations for the *γ* parameter.

• DE dynamics (acceleration of cosmological expansion);

• observational data from gravitational lensing;

• DM dynamics (galactic rotation curves, galaxy cluster masses);

|*γ* − 1|, and from 10−5 for |*β* − 1| (Capozziello, Stabile & Troisi, 2006).

Capozziello, de Filippis & Salzano (2009):

• Tully-Fisher relation.

**Experiment Effect Technique** |*γ* − 1| **lower bound**

On the front of the *f*(*R*) theories, instead, it has been argued that the current estimations of the *γ* and *β* PPN parameters are not sufficient to provide serious constraints on such theories, which claim of being able to explain *observational evidences* about several astrophysical and cosmological problems without any need for Dark Matter (DM) or Dark Energy (DE) like, e.g.

Again, it seems that the desired experimental accuracy can be set at level of 10−7 to 10−8 for

The most stringent experimental limit for *γ* available so far was set by the Cassini experiment (Bertotti, Iess & Tortora, 2003) which used the relativistic time delay effect on the propagation of radar signals (Table 2). This result will be overcome by future astrometric measurements

*α*2 0 0 *α*�

Parametrized Post-Newtonian (PPN) and the Mansouri-Sexl. More details about these two subjects can be found in Will (2006) and in Mansouri & Sexl (1977a;b;c).

## **3.1 The PPN framework**

As pointed out in Will (2006) a viable theory of gravity must be complete, self-consistent, relativistic, and with a correct Post-Newtonian limit. The Post-Newtonian limit of a gravity theory is the approximation of such theory under the assumptions of slow-motion and weak gravity field, i.e. when *v*/*c* ∼ *U*/*c*2 ∼ *�* � 1.3

The PPN formalism is a powerful tool which simplifies the confrontation among the Post-Newtonian limits of the different theories of gravity. This is particularly useful also because these conditions hold in the Solar System, where local experiments are confined.

It can be shown that, in the slow motion and weak-field limit,

$$
\mathfrak{g}\_{a\mathfrak{B}} \simeq \mathfrak{η}\_{a\mathfrak{B}} + h\_{a\mathfrak{B'}} \tag{8}
$$

i.e. that the metric can be written as an expansion about the Minkowskian metric of Special Relativity where the perturbations *hαβ* are functions of the different smallness parameters ∼ *�* like *U*, *v*, etc.

The noticeable thing is that the expansions predicted by nearly every theory of gravity have the same structure apart from the coefficients multiplying each kind of perturbation.

In the PPN formalism these coefficients are replaced by appropriate parameters in such a way that it is possible to reproduce any specific theory simply by assigning specific values to them. Currently the number of parameters of the PPN formalism is set to 10 (Table 1), suitably chosen to give informations on general properties of the selected theory of gravity. On the other side, the formulation of the experimental measurements in the PPN formalism can be translated to the estimation of the values of these parameters, and thus in a direct selection of the viable theories.

The two most important parameters of the PPN formalism are *γ* and *β*, which are both 1 in GR. A precise estimation of these two parameters, has important *theoratical and observational implications*. It could help to *fulfill theoretical needs* because they are the phenomenological "trace" of a scalar field coupled with gravity of scalar-tensor theories which, as explained in the previous section, is related to:

For example, there are formulations of scalar-tensor theories in which the scalar field evolves with time toward a theory close to GR leaving little relic deviations at present times (Damour & Esposito-Farèse, 1992). Such deviations from GR today range from 10−5 to a few times 10−7 for |*γ* − 1|, depending on the cosmological model (Damour & Nordtvedt, 1993).

3 Actually it is also required that the internal energy Π and the pressure-over-density ratio *p*/*ρ* are ∼ *�*.

Table 1. List of the PPN parameters, and of their meanings and values in some theories of gravity.

Table 2. Current best estimations for the *γ* parameter.

On the front of the *f*(*R*) theories, instead, it has been argued that the current estimations of the *γ* and *β* PPN parameters are not sufficient to provide serious constraints on such theories, which claim of being able to explain *observational evidences* about several astrophysical and cosmological problems without any need for Dark Matter (DM) or Dark Energy (DE) like, e.g. Capozziello, de Filippis & Salzano (2009):

4 Will-be-set-by-IN-TECH

Parametrized Post-Newtonian (PPN) and the Mansouri-Sexl. More details about these two

As pointed out in Will (2006) a viable theory of gravity must be complete, self-consistent, relativistic, and with a correct Post-Newtonian limit. The Post-Newtonian limit of a gravity theory is the approximation of such theory under the assumptions of slow-motion and weak

The PPN formalism is a powerful tool which simplifies the confrontation among the Post-Newtonian limits of the different theories of gravity. This is particularly useful also because these conditions hold in the Solar System, where local experiments are confined.

i.e. that the metric can be written as an expansion about the Minkowskian metric of Special Relativity where the perturbations *hαβ* are functions of the different smallness parameters ∼ *�*

The noticeable thing is that the expansions predicted by nearly every theory of gravity have

In the PPN formalism these coefficients are replaced by appropriate parameters in such a way that it is possible to reproduce any specific theory simply by assigning specific values to them. Currently the number of parameters of the PPN formalism is set to 10 (Table 1), suitably chosen to give informations on general properties of the selected theory of gravity. On the other side, the formulation of the experimental measurements in the PPN formalism can be translated to the estimation of the values of these parameters, and thus in a direct

The two most important parameters of the PPN formalism are *γ* and *β*, which are both 1 in GR. A precise estimation of these two parameters, has important *theoratical and observational implications*. It could help to *fulfill theoretical needs* because they are the phenomenological "trace" of a scalar field coupled with gravity of scalar-tensor theories which, as explained in

• theories aiming to provide a formulation of a quantum theory of gravity (Damour, Piazza

For example, there are formulations of scalar-tensor theories in which the scalar field evolves with time toward a theory close to GR leaving little relic deviations at present times (Damour & Esposito-Farèse, 1992). Such deviations from GR today range from 10−5 to a few times 10−7

3 Actually it is also required that the internal energy Π and the pressure-over-density ratio *p*/*ρ* are ∼ *�*.

• theories fully compatible with the Mach principle (Brans & Dicke, 1961); • cosmological scenarios with inflationary stage (Damour & Nordtvedt, 1993);

for |*γ* − 1|, depending on the cosmological model (Damour & Nordtvedt, 1993).

the same structure apart from the coefficients multiplying each kind of perturbation.

*gαβ* � *ηαβ* + *hαβ*, (8)

subjects can be found in Will (2006) and in Mansouri & Sexl (1977a;b;c).

**3.1 The PPN framework**

like *U*, *v*, etc.

selection of the viable theories.

the previous section, is related to:

& Veneziano, 2002).

gravity field, i.e. when *v*/*c* ∼ *U*/*c*2 ∼ *�* � 1.3

It can be shown that, in the slow motion and weak-field limit,

Again, it seems that the desired experimental accuracy can be set at level of 10−7 to 10−8 for |*γ* − 1|, and from 10−5 for |*β* − 1| (Capozziello, Stabile & Troisi, 2006).

The most stringent experimental limit for *γ* available so far was set by the Cassini experiment (Bertotti, Iess & Tortora, 2003) which used the relativistic time delay effect on the propagation of radar signals (Table 2). This result will be overcome by future astrometric measurements

(a) Spherically symmetic case. (b) Deflection pattern with *J*2 � 0.

371 Putting Einstein to Test –

Astrometric Experiments from Space, Fundamental Physics and Local Cosmology

Fig. 1. A non-spherically symmetric distribution of the mass of a perturbing body produces

This asymmetric perturbation on the light path induces specific patterns in the nearby light deflection (Fig. 1) which have never been mesuared up to now because of the smallness of this

Tests of possible violations of the LLI are motivated by several theoretical models encompassing a large number of different subjects, from quantum gravity to varying speed of light cosmologies. A complete review of the tests and of the motivations linked to the LLI is out of scope here, and can be found in Mattingly (2005). Here we will limit ourselves to cite the Robertson-Mansouri-Sexl (RMS) formalism for its possible application to astrometry. A violation of the LLI, in fact, will show itself as a breaking of the Lorentz transformations. The

In analogy to the PPN formalism, the RMS framework is developed under the assumption

depending on a set of arbitrary parameters (*a*, *b*, *d*, ). The potential impact of this formalism on astrometric measurements comes from the fact that those LLI violations depending on *f* = *d*/*b*, show themselves as an aberration effect which therefore puts to test the same properties of the Michelson-Morley experiment. It is probable that violations depending on such aberration effect are out of reach for the present planned space experiments like Gaia (Klioner, 2008) but possible applications to new and more sensitive experiments has not been

**v**(**v** · **x**)

*v*2 + **v** *a t* (11)

RMS formalism is a way to describe this hypothetical breaking in a kinematical way.

that *v* � *c*, and can be expressed as a generalization of the Lorentz transformations

 1 *d* − 1 *b*

*T* = (*t* − · **x**) *a*

**X** = **x** *d* −

specific patterns in the nearby light deflection.

effect.

investigated yet.

Table 3. Current best estimations for the *β* parameter.

from space (Vecchiato et al., 2003). The estimation of *β* depends on that of *γ*, since the former always appear in combination with the latter, and its present bounds are provided by different techniques, involving e.g. Lunar Laser Ranging measurements (Williams, Turyshev & Boggs, 2004) or "Grand fits" of the Solar System equations of motion obtained from the reduction of radar observations (Pitjeva, 2005; 2010) (Table 3). Once again, astrometric techniques could be used to improve on these limits (Vecchiato, Bernardi & Gai, 2010).

### **3.2 Beyond the PPN framework**

Other interesting phenomena which can be used to test the gravity theories at a fundamental level are the higher-order quadrupole contribution to the light deflection, and the possible violations of the Local Lorentz Invariance (LLI). Both can be put to test with convenient astrometric experiments.

The first one is foreseen by GR and other theories of gravity when the light is deflected by perturbing bodies with non-spherically symmetric distributions of the mass. The coefficients of *gαβ*, in fact, in this case does not depend only on the mass, but also on the higher order multipoles of the gravity field of the perturbing body. The deflection can then be described as a vectorial quantity with two components **n** and **m**

$$
\Delta \psi = \Delta \psi\_1 \mathbf{n} + \Delta \psi\_2 \mathbf{m}
$$

with respect to a reference triad (**n**, **m**, **z**) where the **z**-axis is orthogonal to the celestial sphere.

The two components can be modelled as functions of a parameter *ε* whose value is 1 in GR like in the following formulae (Crosta & Mignard, 2006)

$$
\Delta\psi\_1 = \frac{2\left(1+\gamma\right)M}{b} \left[1+\varepsilon l\_2 \frac{R^2}{b^2} \left(1-2\left(\mathbf{n}\cdot\mathbf{z}\right)^2 - \left(\mathbf{t}\cdot\mathbf{z}\right)^2\right)\right] \tag{9}
$$

$$
\Delta\psi\_2 = \frac{4\left(1+\gamma\right)M\varepsilon\|\cdot\|^2\chi}{b^3} \left(\mathbf{m}\cdot\mathbf{z}\right)\left(\mathbf{n}\cdot\mathbf{z}\right) \tag{10}
$$

where *M* is the mass of the perturbing body, *J*2 is the quadrupole component of its gravitational field, and *b* the impact parameter, i.e. the distance of maximum approach of the light path to the perturbing body.

6 Will-be-set-by-IN-TECH

**with** *γ*

excess of Mercury 2*γ* − *β* 3 · 10−3

2*γ* − *β* 1 ÷ 2 · 10−4

**Experiment Effect/Technique Combination**

Perihelion shift

"Grand fits" of Solar System equations of motion

Lunar Laser Ranging Nordtvedt effect 4*β* − *γ* ∼ 10−4

from space (Vecchiato et al., 2003). The estimation of *β* depends on that of *γ*, since the former always appear in combination with the latter, and its present bounds are provided by different techniques, involving e.g. Lunar Laser Ranging measurements (Williams, Turyshev & Boggs, 2004) or "Grand fits" of the Solar System equations of motion obtained from the reduction of radar observations (Pitjeva, 2005; 2010) (Table 3). Once again, astrometric techniques could be

Other interesting phenomena which can be used to test the gravity theories at a fundamental level are the higher-order quadrupole contribution to the light deflection, and the possible violations of the Local Lorentz Invariance (LLI). Both can be put to test with convenient

The first one is foreseen by GR and other theories of gravity when the light is deflected by perturbing bodies with non-spherically symmetric distributions of the mass. The coefficients of *gαβ*, in fact, in this case does not depend only on the mass, but also on the higher order multipoles of the gravity field of the perturbing body. The deflection can then be described as

Δ*ψ* = Δ*ψ*1**n** + Δ*ψ*2**m**

with respect to a reference triad (**n**, **m**, **z**) where the **z**-axis is orthogonal to the celestial sphere. The two components can be modelled as functions of a parameter *ε* whose value is 1 in GR

> *R*2 *b*2

where *M* is the mass of the perturbing body, *J*2 is the quadrupole component of its gravitational field, and *b* the impact parameter, i.e. the distance of maximum approach of

1 − 2 (**n** · **z**)

2 − (**t** · **z**)

*b*3 (**m** · **z**) (**n** · **z**) (10)

2

(9)

 1 + *εJ*2

Radar observations of Mercury (1966-1990)

Radar observations of inner bodies (1963-2003)

**3.2 Beyond the PPN framework**

astrometric experiments.

Table 3. Current best estimations for the *β* parameter.

a vectorial quantity with two components **n** and **m**

like in the following formulae (Crosta & Mignard, 2006)

Δ*ψ*1 = 2 (1 + *γ*) *M b*

the light path to the perturbing body.

Δ*ψ*2 = 4 (1 + *γ*) *MεJ*2*R*2*γ*

used to improve on these limits (Vecchiato, Bernardi & Gai, 2010).

Fig. 1. A non-spherically symmetric distribution of the mass of a perturbing body produces specific patterns in the nearby light deflection.

This asymmetric perturbation on the light path induces specific patterns in the nearby light deflection (Fig. 1) which have never been mesuared up to now because of the smallness of this effect.

Tests of possible violations of the LLI are motivated by several theoretical models encompassing a large number of different subjects, from quantum gravity to varying speed of light cosmologies. A complete review of the tests and of the motivations linked to the LLI is out of scope here, and can be found in Mattingly (2005). Here we will limit ourselves to cite the Robertson-Mansouri-Sexl (RMS) formalism for its possible application to astrometry. A violation of the LLI, in fact, will show itself as a breaking of the Lorentz transformations. The RMS formalism is a way to describe this hypothetical breaking in a kinematical way.

In analogy to the PPN formalism, the RMS framework is developed under the assumption that *v* � *c*, and can be expressed as a generalization of the Lorentz transformations

$$\begin{aligned} T &= \frac{(t - \epsilon \cdot \mathbf{x})}{a} \\ \mathbf{X} &= \frac{\mathbf{x}}{d} - \left(\frac{1}{d} - \frac{1}{b}\right) \frac{\mathbf{v}(\mathbf{v} \cdot \mathbf{x})}{v^2} + \frac{\mathbf{v}}{a} t \end{aligned} \tag{11}$$

depending on a set of arbitrary parameters (*a*, *b*, *d*, ). The potential impact of this formalism on astrometric measurements comes from the fact that those LLI violations depending on *f* = *d*/*b*, show themselves as an aberration effect which therefore puts to test the same properties of the Michelson-Morley experiment. It is probable that violations depending on such aberration effect are out of reach for the present planned space experiments like Gaia (Klioner, 2008) but possible applications to new and more sensitive experiments has not been investigated yet.

Tαβ

represent the space and the time associated to this observer.

2 = −*c*2

time interval measured by the observer *uα*, while similarly d*l*

**4.2 Linking the observable with the unknown parameters**

1 and *kα*

 − 1

d*s*

spatial distance associated to d*s*.

where, as shown in Fig. 4 *kα*

while the four-vectors *kα*

geodesic *xα*, then

can be written as

written as

Γ(τ )

uα

373 Putting Einstein to Test –

Astrometric Experiments from Space, Fundamental Physics and Local Cosmology

Fig. 3. Any time-like curve Γ(*τ*) in the space-time can represent the history of an observer, which is identified by its tangent vector *uα* and whose proper time *τ* is the parameter of the worldline. Locally, *uα* induces a 3 + 1 splitting of the space-time where the two subspaces

*direction* and the *space* relative to the observer *uα* respectively. More precisely, we can always define two operators P*αβ* = −*uαuβ* and T*αβ* = *gαβ* + *uαuβ* which project any four-vector parallely and orthogonally to the observer. Quite obviously, these operators are called the *parallel* and *transverse* projector respectively, and any interval of space-time d*s*2 = *gαβ*d*xα*d*xβ*

> +

> > 1 *k β*

2

T*αβ*d*xα*d*xβ*

2 are the two observing directions (de Felice & Clarke, 1990).

1,2 have the mathematical meaning of the tangents to the path of the

, (13)

(14)

2 = T*αβ*d*xα*d*xβ* is the measured

*c*2 P*αβ*d*xα*d*xβ*

where d*τ*2 = −*c*−2P*αβ*d*xα*d*xβ* has the physical meaning of the (square of the) infinitesimal

With the above indications in mind, and remembering the spatial nature of angular measurements, it is easy to understand why the relativistic counterpart of Eq. (12) can be

> T*αβkα* 1 *k β* 1 T*αβkα* 2 *k β* 2

It has to be noticed, however, that the expression "observing direction" has not to be intended as the result of a measure. The observable is the angle at the left-hand-side of the equation,

incoming light rays connecting the positions of the observed objects to the observer, i.e. of the null geodesics of the observed photons. In other words, if *s* is the parameter of the null

> *kα* = d*xα* d*s*

cos *ψ*12 = T*αβkα*

## **4. Modelling the observations**

#### **4.1 Basics of the relativistic Theory of measurements**

In order to understand why Astrometry can be used to set limits to the PPN parameters and therefore to put to test the different theories of gravity, it has to be shown how these parameters enter in the astrometric observable. The most basic measure in Astrometry, ideally, is the angle *ψ*12 between two observing directions, which in the usual Euclidean geometry reads

$$\cos\psi\_{12} = \frac{\mathbf{r}\_1 \cdot \mathbf{r}\_2}{|\mathbf{r}\_1| \, |\mathbf{r}\_2|} \, \tag{12}$$

and can be represented as in Fig. 2 as the projection on a unit sphere of the two vectors **r**1 and **r**2 connecting the observer and the objects P and Q.

Fig. 2. Representation of the angular distance *ψ* on the unit sphere between two objects P and Q.

In order to give a correct interpretation of the experimental results, however, the measurements must be written in a proper relativistic way, i.e. following the prescription of the *theory of measurements*. The details of this formalism are out of scope in this article, and are fully developed in de Felice & Bini (2010). We therefore will give here only a brief overview of the main concepts needed to follow the present exposition.

The starting point is the definition of a physical observer, whose history, as shown in Fig. 3, is identified by his/her worldline Γ(*w*) in the four-dimensional differential manifold representing the space-time. The four-vector tangent to this curve (i.e. the *four-velocity* of Γ) is *uα* ≡ Γ˙ , where the dot stands for the derivative with respect to the parameter *w* of the worldline, and if the four-velocity is unitary with respect to the metric *gαβ* of the space-time, that is if *gαβuαuβ* = −1, then *uα* represents the observer at a given point of the worldline. The parameter *τ* of the curve Γ that satisfies the unitarity condition of *uα* has the physical meaning of the *proper time* of the observer, so that *uα* can be interpreted as the representation of the observer at each instant of his/her proper time.

With a certain degree of approximation, it can be said that at each point of Γ, *uα* splits the space-time into the 1D and 3D subspaces parallel and orthogonal to itself, which are the *time* 8 Will-be-set-by-IN-TECH

In order to understand why Astrometry can be used to set limits to the PPN parameters and therefore to put to test the different theories of gravity, it has to be shown how these parameters enter in the astrometric observable. The most basic measure in Astrometry, ideally, is the angle *ψ*12 between two observing directions, which in the usual Euclidean geometry

cos *ψ*12 = **r**1 · **r**2

and can be represented as in Fig. 2 as the projection on a unit sphere of the two vectors **r**1 and

Fig. 2. Representation of the angular distance *ψ* on the unit sphere between two objects P and

In order to give a correct interpretation of the experimental results, however, the measurements must be written in a proper relativistic way, i.e. following the prescription of the *theory of measurements*. The details of this formalism are out of scope in this article, and are fully developed in de Felice & Bini (2010). We therefore will give here only a brief

The starting point is the definition of a physical observer, whose history, as shown in Fig. 3, is identified by his/her worldline Γ(*w*) in the four-dimensional differential manifold representing the space-time. The four-vector tangent to this curve (i.e. the *four-velocity* of Γ) is *uα* ≡ Γ˙ , where the dot stands for the derivative with respect to the parameter *w* of the worldline, and if the four-velocity is unitary with respect to the metric *gαβ* of the space-time, that is if *gαβuαuβ* = −1, then *uα* represents the observer at a given point of the worldline. The parameter *τ* of the curve Γ that satisfies the unitarity condition of *uα* has the physical meaning of the *proper time* of the observer, so that *uα* can be interpreted as the representation of the

With a certain degree of approximation, it can be said that at each point of Γ, *uα* splits the space-time into the 1D and 3D subspaces parallel and orthogonal to itself, which are the *time*

overview of the main concepts needed to follow the present exposition.

observer at each instant of his/her proper time.

, (12)

**4. Modelling the observations**

reads

Q.

**4.1 Basics of the relativistic Theory of measurements**

**r**2 connecting the observer and the objects P and Q.

Fig. 3. Any time-like curve Γ(*τ*) in the space-time can represent the history of an observer, which is identified by its tangent vector *uα* and whose proper time *τ* is the parameter of the worldline. Locally, *uα* induces a 3 + 1 splitting of the space-time where the two subspaces represent the space and the time associated to this observer.

*direction* and the *space* relative to the observer *uα* respectively. More precisely, we can always define two operators P*αβ* = −*uαuβ* and T*αβ* = *gαβ* + *uαuβ* which project any four-vector parallely and orthogonally to the observer. Quite obviously, these operators are called the *parallel* and *transverse* projector respectively, and any interval of space-time d*s*2 = *gαβ*d*xα*d*xβ* can be written as

$$\mathbf{ds}^2 = -c^2 \left( -\frac{1}{c^2} \mathcal{P}\_{a\beta} \mathbf{dx}^a \mathbf{dx}^\beta \right) + \left( \mathcal{T}\_{a\beta} \mathbf{dx}^a \mathbf{dx}^\beta \right)\_{\prime} \tag{13}$$

where d*τ*2 = −*c*−2P*αβ*d*xα*d*xβ* has the physical meaning of the (square of the) infinitesimal time interval measured by the observer *uα*, while similarly d*l* 2 = T*αβ*d*xα*d*xβ* is the measured spatial distance associated to d*s*.

#### **4.2 Linking the observable with the unknown parameters**

With the above indications in mind, and remembering the spatial nature of angular measurements, it is easy to understand why the relativistic counterpart of Eq. (12) can be written as

$$\cos\psi\_{12} = \frac{\mathcal{T}\_{a\beta}k\_1^a k\_2^\beta}{\sqrt{\mathcal{T}\_{a\beta}k\_1^a k\_1^\beta}\sqrt{\mathcal{T}\_{a\beta}k\_2^a k\_2^\beta}}\tag{14}$$

where, as shown in Fig. 4 *kα* 1 and *kα* 2 are the two observing directions (de Felice & Clarke, 1990). It has to be noticed, however, that the expression "observing direction" has not to be intended as the result of a measure. The observable is the angle at the left-hand-side of the equation, while the four-vectors *kα* 1,2 have the mathematical meaning of the tangents to the path of the incoming light rays connecting the positions of the observed objects to the observer, i.e. of the null geodesics of the observed photons. In other words, if *s* is the parameter of the null geodesic *xα*, then

$$k^{\alpha} = \frac{\mathbf{d}x^{\alpha}}{\mathbf{ds}}$$

model and/or to the specific technique used for the integration of the equation of motion.

375 Putting Einstein to Test –

Astrometric Experiments from Space, Fundamental Physics and Local Cosmology

gravitational influence of all the bodies of the Solar System. More accurate sub-*μ*as models can be found in Klioner (2003) and references therein, Kopeikin & Schäfer (1999), and de Felice et al. (2006). Finally, Teyssandier & Le Poncin-Lafitte (2008) have applied the method of the Synge's World Function to obtain a relativistic expression of the basic astrometric observables

Whatever is the explicit method used for the integration of the null geodesic, the PPN parameters such as *γ* and *β* can be easily included in the formula of the observable by chosing an appropriate PPN expression of the metric tensor, however there exists an important difference between them. If we consider, e.g., the PPN approximation of the Schwarzschild metric4 we can see that its expression in isotropic coordinates at the first Post-Newtonian order

> d*t*

This shows that *β* enter the metric at the second order in the gravitational potential *U*, while

this explains why astrometric measurements exploiting the light deflection are particularly suited for the estimation of *γ* but not for that of *β*. The latter, actually, can also be addressed by astrometric measurements able to detect the well-known effect of the excess of perihelion

in the orbit of massive objects. If we neglect the contribution from the other PPN parameters and from the quadrupole of the gravitational source, which in the case of the Sun is not important, we can see that this observation is affected by *γ* and *β* at the same order, and therefore *β* can be estimated given a prior knowledge of *γ*, usually known at a better order

If the required measure is the observed direction of the object, instead of an angle between two directions, then one has to deal with a relativistic theory of reference systems. The Equivalence Principle states that for any observer it is always possible to find a locally inertial reference system where Special Relativity holds. This reference system is called *tetrad*, and it

4 The Schwarzschild metric considers the spherical non-rotating Sun as the only source of gravity.

−1 , *gμν*

3 (2 + 2*γ* − *β*) + *f* (*α*1, *α*2, *α*3, *ζ*2, *J*2)

*Eα* 0ˆ , *Eα* 1ˆ , *Eα* 2ˆ , *Eα* 3ˆ

 (*g*00)

1 − 2*U* + 2*βU*2

*γ* is at the first order, and since the relativistic deflection of light is in general

Δ*ψ* = *f*

 1

2 model is described in Vecchiato et al. (2003) and reference therein, while

2 + (1 + 2*γU*) *δij*d*xi*

2 model which takes into account the

d*xj*

, (17)

. (16)

which can be built at

(18)

A sub-mas (*v*/*c*)

(1PN) can be written as

shift

de Felice et al. (2004) is an example of a *μ*as-level (*v*/*c*)

without integrating directly the geodesic equations.

d*s* 2 = −

Δ*ω* = 6*πm*

**4.3 Relativistic reference systems in astrometry**

is represented by a set of 4 orthonormal four-vectors

because of the above considerations.

*a* (1 − *e*2)

Fig. 4. The formula expressing the arc measured between two directions can be found considering that the relativistic viewing directions *kα* 1 and *kα* 2 – i.e. the tangents to the null geodesics connecting the observed objects to the observer and computed at the observation event – have to be projected onto the *spatial* hypersurface relative to *uα*.

and this four-vector can be obtained by solving the geodesic equation (i.e. the relativistic equation of motion)

$$
\frac{\mathrm{d}^2 \mathrm{x}^\alpha}{\mathrm{d}s^2} + \Gamma^\alpha\_{\beta\gamma} \frac{\mathrm{d} \mathrm{x}^\beta}{\mathrm{d}s} \frac{\mathrm{d} \mathrm{x}^\gamma}{\mathrm{d}s} = 0.\tag{15}
$$

This formula has to be expressed as a function of the desired astrometric unknowns, namely the positions and proper motions of the observed objects. Moreover, as it is known from classical astrometry (see e.g. Smart, 1965), measurements of the same object change in time not only because of its intrinsic motion, but also because of the motion of the observer itself, with the aberration and parallax effects. In addition to this, a relativistic formulation of the astrometric measurements has to properly take into account the geometry of the space-time.

The most important thing to be noticed in Eq. (14) is that it is natural to choose a coordinate system of the space-time metric coincident with that of the desired catalog, and that this helps to express automatically the observer as a function of the needed parameters introducing immediately and naturally the observer-related effects of classical astrometry like the aberration and the parallactic displacement. If we imagine, just as an example, to use the Schwarzschild metric, its coordinates are already centered on the Sun, therefore the aberration is automatically taken into account in the observation equation since the transverse projector

$$\mathcal{T}\_{\alpha\beta} = \mathcal{g}\_{\alpha\beta} + \mu\_{\alpha}\mu\_{\beta}$$

contains the (four-)velocity *uα* expressed with respect to the origin of the coordinate system.

It's easy to understand that the dependence on the positions of the observed objects enters in the equation through the *kα i* , while the relativistic formulation of the observable, together with the correct solution of Eq. (15), can take into account of all the "relativistic effects" such as the light deflection, i.e. the geometry of the space-time. However, many ways to address the problem of the integration of the null geodesics have been developed so far, therefore the actual expression of Eq. (14) can vary according to the accuracy level of the 10 Will-be-set-by-IN-TECH

uα

Fig. 4. The formula expressing the arc measured between two directions can be found

event – have to be projected onto the *spatial* hypersurface relative to *uα*.

d2*xα* d*s*2 + Γ*α*

geodesics connecting the observed objects to the observer and computed at the observation

and this four-vector can be obtained by solving the geodesic equation (i.e. the relativistic

This formula has to be expressed as a function of the desired astrometric unknowns, namely the positions and proper motions of the observed objects. Moreover, as it is known from classical astrometry (see e.g. Smart, 1965), measurements of the same object change in time not only because of its intrinsic motion, but also because of the motion of the observer itself, with the aberration and parallax effects. In addition to this, a relativistic formulation of the astrometric measurements has to properly take into account the geometry of the space-time. The most important thing to be noticed in Eq. (14) is that it is natural to choose a coordinate system of the space-time metric coincident with that of the desired catalog, and that this helps to express automatically the observer as a function of the needed parameters introducing immediately and naturally the observer-related effects of classical astrometry like the aberration and the parallactic displacement. If we imagine, just as an example, to use the Schwarzschild metric, its coordinates are already centered on the Sun, therefore the aberration is automatically taken into account in the observation equation since the transverse projector

T*αβ* = *gαβ* + *uαuβ* contains the (four-)velocity *uα* expressed with respect to the origin of the coordinate system. It's easy to understand that the dependence on the positions of the observed objects enters

with the correct solution of Eq. (15), can take into account of all the "relativistic effects" such as the light deflection, i.e. the geometry of the space-time. However, many ways to address the problem of the integration of the null geodesics have been developed so far, therefore the actual expression of Eq. (14) can vary according to the accuracy level of the

*i* , while the relativistic formulation of the observable, together

d*xγ*

*βγ* d*xβ* d*s*

Γ(s)

1 and *kα*

2 – i.e. the tangents to the null

d*s* = 0. (15)

Tαβ

considering that the relativistic viewing directions *kα*

equation of motion)

in the equation through the *kα*

model and/or to the specific technique used for the integration of the equation of motion. A sub-mas (*v*/*c*) 2 model is described in Vecchiato et al. (2003) and reference therein, while de Felice et al. (2004) is an example of a *μ*as-level (*v*/*c*) 2 model which takes into account the gravitational influence of all the bodies of the Solar System. More accurate sub-*μ*as models can be found in Klioner (2003) and references therein, Kopeikin & Schäfer (1999), and de Felice et al. (2006). Finally, Teyssandier & Le Poncin-Lafitte (2008) have applied the method of the Synge's World Function to obtain a relativistic expression of the basic astrometric observables without integrating directly the geodesic equations.

Whatever is the explicit method used for the integration of the null geodesic, the PPN parameters such as *γ* and *β* can be easily included in the formula of the observable by chosing an appropriate PPN expression of the metric tensor, however there exists an important difference between them. If we consider, e.g., the PPN approximation of the Schwarzschild metric4 we can see that its expression in isotropic coordinates at the first Post-Newtonian order (1PN) can be written as

$$\mathbf{ds}^2 = -\left(1 - 2\mathcal{U} + 2\beta\mathcal{U}^2\right)\mathbf{d}t^2 + \left(1 + 2\gamma\mathcal{U}\right)\delta\_{\vec{l}\vec{l}}\mathbf{dx}^i\mathbf{dx}^j.\tag{16}$$

This shows that *β* enter the metric at the second order in the gravitational potential *U*, while *γ* is at the first order, and since the relativistic deflection of light is in general

$$